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Theorem List for Metamath Proof Explorer - 28201-28300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxccosh 28201 Extend class notation to include the hyperbolic cosine function. see df-cosh 28204.
 class cosh
 
Syntaxctanh 28202 Extend class notation to include the hyperbolic tangent function, see df-tanh 28205.
 class tanh
 
Definitiondf-sinh 28203 Define the hyperbolic sine function (sinh). We define it this way for cmpt 4077, which requires the form  (
x  e.  A  |->  B ). See sinhval-named 28206 for a simple way to evaluate it. We define this function by dividing by  _i, which uses fewer operations than many conventional definitions (and thus is more convenient to use in metamath). See sinh-conventional 28209 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)
 |- sinh  =  ( x  e.  CC  |->  ( ( sin `  ( _i  x.  x ) ) 
 /  _i ) )
 
Definitiondf-cosh 28204 Define the hyperbolic cosine function (cosh). We define it this way for cmpt 4077, which requires the form  (
x  e.  A  |->  B ). (Contributed by David A. Wheeler, 10-May-2015.)
 |- cosh  =  ( x  e.  CC  |->  ( cos `  ( _i  x.  x ) ) )
 
Definitiondf-tanh 28205 Define the hyperbolic tangent function (tanh). We define it this way for cmpt 4077, which requires the form  (
x  e.  A  |->  B ). (Contributed by David A. Wheeler, 10-May-2015.)
 |- tanh  =  ( x  e.  ( `'cosh " ( CC  \  { 0 } )
 )  |->  ( ( tan `  ( _i  x.  x ) )  /  _i ) )
 
Theoremsinhval-named 28206 Value of the named sinh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-sinh 28203. See sinhval 12434 for a theorem to convert this further. See sinh-conventional 28209 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)
 |-  ( A  e.  CC  ->  (sinh `  A )  =  ( ( sin `  ( _i  x.  A ) ) 
 /  _i ) )
 
Theoremcoshval-named 28207 Value of the named cosh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-cosh 28204. See coshval 12435 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  CC  ->  (cosh `  A )  =  ( cos `  ( _i  x.  A ) ) )
 
Theoremtanhval-named 28208 Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh 28205. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  ( `'cosh " ( CC  \  {
 0 } ) ) 
 ->  (tanh `  A )  =  ( ( tan `  ( _i  x.  A ) ) 
 /  _i ) )
 
Theoremsinh-conventional 28209 Conventional definition of sinh. Here we show that the sinh definition we're using has the same meaning as the conventional definition used in some other sources. We choose a slightly different definition of sinh because it has fewer operations, and thus is more convenient to manipulate using metamath. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  CC  ->  (sinh `  A )  =  (
 -u _i  x.  ( sin `  ( _i  x.  A ) ) ) )
 
Theoremsinhpcosh 28210 Prove that  (sinh `  A
)  +  (cosh `  A )  =  ( exp `  A ) using the conventional hyperbolic trig functions. (Contributed by David A. Wheeler, 27-May-2015.)
 |-  ( A  e.  CC  ->  ( (sinh `  A )  +  (cosh `  A )
 )  =  ( exp `  A ) )
 
18.24.4  Reciprocal trig functions (sec, csc, cot)

Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them.

 
Syntaxcsec 28211 Extend class notation to include the secant function, see df-sec 28214.
 class  sec
 
Syntaxccsc 28212 Extend class notation to include the cosecant function, see df-csc 28215.
 class  csc
 
Syntaxccot 28213 Extend class notation to include the cotangent function, see df-cot 28216.
 class  cot
 
Definitiondf-sec 28214* Define the secant function. We define it this way for cmpt 4077, which requires the form  ( x  e.  A  |->  B ). The sec function is defined in ISO 80000-2:2009(E) operation 2-13.6 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  sec  =  ( x  e.  {
 y  e.  CC  |  ( cos `  y )  =/=  0 }  |->  ( 1 
 /  ( cos `  x ) ) )
 
Definitiondf-csc 28215* Define the cosecant function. We define it this way for cmpt 4077, which requires the form  ( x  e.  A  |->  B ). The csc function is defined in ISO 80000-2:2009(E) operation 2-13.7 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  csc  =  ( x  e.  {
 y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( 1 
 /  ( sin `  x ) ) )
 
Definitiondf-cot 28216* Define the cotangent function. We define it this way for cmpt 4077, which requires the form  ( x  e.  A  |->  B ). The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  cot  =  ( x  e.  {
 y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( ( cos `  x )  /  ( sin `  x ) ) )
 
Theoremsecval 28217 Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  =  ( 1  /  ( cos `  A ) ) )
 
Theoremcscval 28218 Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  =  ( 1  /  ( sin `  A ) ) )
 
Theoremcotval 28219 Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  =  ( ( cos `  A )  /  ( sin `  A ) ) )
 
Theoremseccl 28220 The closure of the secant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  e.  CC )
 
Theoremcsccl 28221 The closure of the cosecant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  e.  CC )
 
Theoremcotcl 28222 The closure of the cotangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  e.  CC )
 
Theoremreseccl 28223 The closure of the secant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  e.  RR )
 
Theoremrecsccl 28224 The closure of the cosecant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  e.  RR )
 
Theoremrecotcl 28225 The closure of the cotangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  e.  RR )
 
Theoremrecsec 28226 The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( cos `  A )  =  ( 1  /  ( sec `  A ) ) )
 
Theoremreccsc 28227 The reciprocal of cosecant is sine. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( sin `  A )  =  ( 1  /  ( csc `  A ) ) )
 
Theoremreccot 28228 The reciprocal of cotangent is tangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0  /\  ( cos `  A )  =/=  0 )  ->  ( tan `  A )  =  ( 1  /  ( cot `  A ) ) )
 
Theoremrectan 28229 The reciprocal of tangent is cotangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0  /\  ( cos `  A )  =/=  0 )  ->  ( cot `  A )  =  ( 1  /  ( tan `  A ) ) )
 
Theoremsec0 28230 The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( sec `  0 )  =  1
 
Theoremonetansqsecsq 28231 Prove the tangent squared secant squared identity  ( 1  +  ( ( tan A ) ^ 2 ) ) = ( ( sec  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 25-May-2015.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( 1  +  (
 ( tan `  A ) ^ 2 ) )  =  ( ( sec `  A ) ^ 2
 ) )
 
Theoremcotsqcscsq 28232 Prove the tangent squared cosecant squared identity  ( 1  +  ( ( cot A ) ^ 2 ) ) = ( ( csc  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 27-May-2015.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( 1  +  (
 ( cot `  A ) ^ 2 ) )  =  ( ( csc `  A ) ^ 2
 ) )
 
18.24.5  Identities for "if"

Utility theorems for "if".

 
Theoremifnmfalse 28233 If A is not a member of B, but an "if" condition requires it, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3572 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e/  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
 
18.24.6  Not-member-of
 
TheoremAnelBC 28234 If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using 
e/. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  A  =/=  B   &    |-  A  =/=  C   =>    |-  A  e/  { B ,  C }
 
18.24.7  Decimal point

Define the decimal point operator and the decimal fraction constructor. This can model traditional decimal point notation, and serve as a convenient way to write some fractional numbers. See df-dp 28238 and df-dp2 28237 for more information; ~? dfpval provides a more convenient way to obtain a value. This is intentionally similar to df-dec 10125.

TODO: Fix non-existent label dfpval.

 
Syntaxcdp2 28235 Constant used for decimal fraction constructor. See df-dp2 28237.
 class _ A B
 
Syntaxcdp 28236 Decimal point operator. See df-dp 28238.
 class  period
 
Definitiondf-dp2 28237 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 10125. (Contributed by David A. Wheeler, 15-May-2015.)
 |- _ A B  =  ( A  +  ( B 
 /  10 ) )
 
Definitiondf-dp 28238* Define the  period (decimal point) operator. For example,  ( 1 period 5 )  =  ( 3  /  2 ), and  -u (; 3 2 period_ 7_ 1 8 )  = 
-u (;;;; 3 2 7 1 8  / ;;; 1 0 0 0 ) Unary minus, if applied, should normally be applied in front of the parentheses.

Metamath intentionally does not have a built-in construct for numbers, so it can show that numbers are something you can build based on set theory. However, that means that metamath has no built-in way to handle decimal numbers as traditionally written, e.g., "2.54", and its parsing system intentionally does not include the complexities necessary to define such a parsing system. Here we create a system for modeling traditional decimal point notation; it is not syntactically identical, but it is sufficiently similar so it is a reasonable model of decimal point notation. It should also serve as a convenient way to write some fractional numbers.

The RHS is  RR, not  QQ; this should simplify some proofs. The LHS is  NN0, since that is what is used in practice. The definition intentionally does not allow negative numbers on the LHS; if it did, nonzero fractions would produce the wrong results. (It would be possible to define the decimal point to do this, but using it would be more complicated, and the expression  -u ( A period B ) is just as convenient.) (Contributed by David A. Wheeler, 15-May-2015.)

 |-  period  =  ( x  e.  NN0 ,  y  e.  RR  |-> _ x y )
 
Theoremdp2cl 28239 Define the closure for the decimal fraction constructor if both values are reals. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  -> _ A B  e.  RR )
 
Theoremdpval 28240 Define the value of the decimal point operator. See df-dp 28238. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  = _ A B )
 
Theoremdpcl 28241 Prove that the closure of the decimal point is  RR as we have defined it. See df-dp 28238. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  e.  RR )
 
Theoremdpfrac1 28242 Prove a simple equivalence involving the decimal point. See df-dp 28238 and dpcl 28241. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  =  (; A B  /  10 ) )
 
18.24.8  Signum (sgn or sign) function
 
Syntaxcsgn 28243 Extend class notation to include the Signum function.
 class sgn
 
Definitiondf-sgn 28244 Signum function. Pronounced "signum" , otherwise it might be confused with "sine". Defined as "sgn" in ISO 80000-2:2009(E) operation 2-9.13. It is named "sign" (with the same definition) in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. We define this over  RR* (df-xr 8871) instead of  RR so that it can accept  +oo and  -oo. Note that df-psgn 27415 defines the sign of a permutation, which is different. Value shown in sgnval 28245. (Contributed by David A. Wheeler, 15-May-2015.)
 |- sgn  =  ( x  e.  RR*  |->  if ( x  =  0 , 
 0 ,  if ( x  <  0 ,  -u 1 ,  1 ) ) )
 
Theoremsgnval 28245 Value of Signum function. Pronounced "signum" . See df-sgn 28244. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
 
Theoremsgn0 28246 Proof that signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (sgn `  0 )  =  0
 
Theoremsgnp 28247 Proof that signum of positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR*  /\  0  <  A ) 
 ->  (sgn `  A )  =  1 )
 
Theoremsgnrrp 28248 Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.)
 |-  ( A  e.  RR+  ->  (sgn `  A )  =  1 )
 
Theoremsgn1 28249 Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn `  1 )  =  1
 
Theoremsgnpnf 28250 Proof that the signum of  +oo is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn ` 
 +oo )  =  1
 
Theoremsgnn 28251 Proof that signum of negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR*  /\  A  <  0 ) 
 ->  (sgn `  A )  =  -u 1 )
 
Theoremsgnmnf 28252 Proof that the signum of  -oo is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn ` 
 -oo )  =  -u 1
 
18.24.9  Ceiling function
 
Syntaxccei 28253 Extend class notation to include the ceiling function.
 class
 
Definitiondf-ceiling 28254 The ceiling function. Defined in ISO 80000-2:2009(E) operation 2-9.18 and the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4.

By convention metamath users tend to expand this construct directly, instead of using the definition. However, we want to make sure this is separately and formally defined. Proof ceicl 10955 shows that the ceiling function returns an integer when provided a real. Formalized by David A. Wheeler. (Contributed by David A. Wheeler, 19-May-2015.)

 |- =  ( x  e.  RR  |->  -u ( |_ `  -u x ) )
 
Theoremceilingval 28255 The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
 |-  ( A  e.  RR  ->  ( `  A )  =  -u ( |_ `  -u A ) )
 
Theoremceilingcl 28256 Closure of the ceiling function; the real work is in ceicl 10955. (Contributed by David A. Wheeler, 19-May-2015.)
 |-  ( A  e.  RR  ->  ( `  A )  e.  ZZ )
 
18.24.10  Logarithm laws generalized to an arbitrary base - log_

Define "log using an arbitrary base" function and then prove some of its properties. This builds on previous work by Stefan O'Rear.

This supports the notational form  ( (logb `  B ) `  X
); that looks a little more like traditional notation, but is different than other 2-parameter functions. E.G.,  ( (log_ `  10 ) ` ;; 1 0 0 )  =  2

 
Syntaxclog_ 28257 Extend class notation to include the logarithm generalized to an arbitrary base.
 class log_
 
Definitiondf-log_ 28258* Define the log_ operator. This is the logarithm generalized to an arbitrary base. It can be used as  ( (log_ `  B ) `  X ) for "log base B of X". This formulation suggested by Mario Carneiro. (Contributed by David A. Wheeler, 14-Jul-2017.)
 |- log_  =  ( b  e.  ( CC  \  { 0 ,  1 } )  |->  ( x  e.  ( CC  \  { 0 } )  |->  ( ( log `  x )  /  ( log `  b
 ) ) ) )
 
18.24.11  Miscellaneous

Miscellaneous proofs.

 
Theorem5m4e1 28259 Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.)
 |-  (
 5  -  4 )  =  1
 
Theorem2p2ne5 28260 Prove that  2  +  2  =/=  5. In George Orwell's "1984", Part One, Chapter Seven, the protagonist Winston notes that, "In the end the Party would announce that two and two made five, and you would have to believe it." http://www.sparknotes.com/lit/1984/section4.rhtml. More generally, the phrase  2  +  2  =  5 has come to represent an obviously false dogma one may be required to believe. See the Wikipedia article for more about this: https://en.wikipedia.org/wiki/2_%2B_2_%3D_5. Unsurprisingly, we can easily prove that this claim is false. (Contributed by David A. Wheeler, 31-Jan-2017.)
 |-  (
 2  +  2 )  =/=  5
 
Theoremresolution 28261 Resolution rule. This is the primary inference rule in some automated theorem provers such as prover9. The resolution rule can be traced back to Davis and Putnam (1960). (Contributed by David A. Wheeler, 9-Feb-2017.)
 |-  (
 ( ( ph  /\  ps )  \/  ( -.  ph  /\ 
 ch ) )  ->  ( ps  \/  ch )
 )
 
18.25  Mathbox for Alan Sare
 
18.25.1  Conventional Metamath proofs, some derived from VD proofs
 
Theoremiidn3 28262 idn3 28387 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ch )
 ) )
 
Theoremee222 28263 e222 28408 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th )
 )   &    |-  ( ph  ->  ( ps  ->  ta ) )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoremee3bir 28264 Right-biconditional form of e3 28512 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ta  <->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
 
Theoremee13 28265 e13 28523 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  ( th  ->  ta ) ) )   &    |-  ( ps  ->  ( ta  ->  et ) )   =>    |-  ( ph  ->  ( ch  ->  ( th  ->  et ) ) )
 
Theoremee121 28266 e121 28428 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   &    |-  ( ph  ->  ta )   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ch  ->  et ) )
 
Theoremee122 28267 e122 28425 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   &    |-  ( ph  ->  ( ch  ->  ta ) )   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ch  ->  et ) )
 
Theoremee333 28268 e333 28508 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  et )
 ) )   &    |-  ( th  ->  ( ta  ->  ( et  ->  ze ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ze )
 ) )
 
Theoremee323 28269 e323 28541 without virtual deductions. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ( ps  ->  ta )
 )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )   &    |-  ( th  ->  ( ta  ->  ( et  ->  ze )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ze ) ) )
 
Theorem3ornot23 28270 If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 28623. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( -.  ph  /\  -.  ps )  ->  ( ( ch  \/  ph  \/  ps )  ->  ch ) )
 
Theoremorbi1r 28271 orbi1 686 with order of disjuncts reversed. Derived from orbi1rVD 28624. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  \/  ph ) 
 <->  ( ch  \/  ps ) ) )
 
Theorembitr3 28272 Closed nested implication form of bitr3i 242. Derived automatically from bitr3VD 28625. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ph  <->  ch )  ->  ( ps 
 <->  ch ) ) )
 
Theorem3orbi123 28273 pm4.39 841 with a 3-conjunct antecedent. This proof is 3orbi123VD 28626 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th )  /\  ( ta 
 <->  et ) )  ->  ( ( ph  \/  ch 
 \/  ta )  <->  ( ps  \/  th 
 \/  et ) ) )
 
Theoremsyl5imp 28274 Closed form of syl5 28. Derived automatically from syl5impVD 28639. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ( th  ->  ps )  ->  ( ph  ->  ( th  ->  ch )
 ) ) )
 
Theoremimpexp3a 28275 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. After the User's Proof was completed it was minimized. The completed User's Proof before minimization is not shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
1::  |-  ( ( ( ps  /\  ch )  ->  th )  <->  ( ps  ->  ( ch  ->  th ) ) )
qed:1:  |-  ( ( ph  ->  ( ( ps  /\  ch )  ->  th ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
 |-  (
 ( ph  ->  ( ( ps  /\  ch )  ->  th ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
 
Theoremcom3rgbi 28276 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
1::  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->  ( ph  ->  ( ch  ->  ( ps  ->  th ) ) ) )
2::  |-  ( ( ph  ->  ( ch  ->  ( ps  ->  th ) ) )  ->  ( ch  ->  ( ph  ->  ( ps  ->  th ) ) ) )
3:1,2:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->  ( ch  ->  ( ph  ->  ( ps  ->  th ) ) ) )
4::  |-  ( ( ch  ->  ( ph  ->  ( ps  ->  th ) ) )  ->  ( ph  ->  ( ch  ->  ( ps  ->  th ) ) ) )
5::  |-  ( ( ph  ->  ( ch  ->  ( ps  ->  th ) ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
6:4,5:  |-  ( ( ch  ->  ( ph  ->  ( ps  ->  th ) ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
qed:3,6:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  <->  ( ch  ->  ( ph  ->  ( ps  ->  th ) ) ) )
 |-  (
 ( ph  ->  ( ps 
 ->  ( ch  ->  th )
 ) )  <->  ( ch  ->  (
 ph  ->  ( ps  ->  th ) ) ) )
 
Theoremimpexp3acom3r 28277 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
1::  |-  ( ( ph  ->  ( ( ps  /\  ch )  ->  th ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
2::  |-  ( ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
qed:1,2:  |-  ( ( ph  ->  ( ( ps  /\  ch )  ->  th ) )  <->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
 |-  (
 ( ph  ->  ( ( ps  /\  ch )  ->  th ) )  <->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
 
Theoremee1111 28278 Non-virtual deduction form of e1111 28447. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
h1::  |-  ( ph  ->  ps )
h2::  |-  ( ph  ->  ch )
h3::  |-  ( ph  ->  th )
h4::  |-  ( ph  ->  ta )
h5::  |-  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) )
6:1,5:  |-  ( ph  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) )
7:6:  |-  ( ch  ->  ( ph  ->  ( th  ->  ( ta  ->  et ) ) ) )
8:2,7:  |-  ( ph  ->  ( ph  ->  ( th  ->  ( ta  ->  et ) ) ) )
9:8:  |-  ( ph  ->  ( th  ->  ( ta  ->  et ) ) )
10:9:  |-  ( th  ->  ( ph  ->  ( ta  ->  et ) ) )
11:3,10:  |-  ( ph  ->  ( ph  ->  ( ta  ->  et ) ) )
12:11:  |-  ( ph  ->  ( ta  ->  et ) )
13:12:  |-  ( ta  ->  ( ph  ->  et ) )
14:4,13:  |-  ( ph  ->  ( ph  ->  et ) )
qed:14:  |-  ( ph  ->  et )
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) )   =>    |-  ( ph  ->  et )
 
Theorempm2.43bgbi 28279 Logical equivalence of a 2-left-nested implication and a 1-left-nested implicated when two antecedents of the former implication are identical. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
1::  |-  ( ( ph  ->  ( ps  ->  ( ph  ->  ch ) ) )  ->  ( ph  ->  ( ph  ->  ( ps  ->  ch ) ) ) )
2::  |-  ( ( ph  ->  ( ph  ->  ( ps  ->  ch ) ) )  ->  ( ph  ->  ( ps  ->  ch ) ) )
3:1,2:  |-  ( ( ph  ->  ( ps  ->  ( ph  ->  ch ) ) )  ->  ( ph  ->  ( ps  ->  ch ) ) )
4::  |-  ( ( ph  ->  ( ps  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )
5:3,4:  |-  ( ( ph  ->  ( ps  ->  ( ph  ->  ch ) ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )
6::  |-  ( ( ps  ->  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  ->  ( ph  ->  ch ) ) ) )
qed:5,6:  |-  ( ( ph  ->  ( ps  ->  ( ph  ->  ch ) ) )  <->  ( ps  ->  ( ph  ->  ch ) ) )
 |-  (
 ( ph  ->  ( ps 
 ->  ( ph  ->  ch )
 ) )  <->  ( ps  ->  (
 ph  ->  ch ) ) )
 
Theorempm2.43cbi 28280 Logical equivalence of a 3-left-nested implication and a 2-left-nested implicated when two antecedents of the former implication are identical. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
1::  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  )  ->  ( ph  ->  ( ps  ->  ( ph  ->  ( ch  ->  th ) ) ) ) )
2::  |-  ( ( ph  ->  ( ps  ->  ( ph  ->  ( ch  ->  th ) ) )  )  ->  ( ps  ->  ( ph  ->  ( ch  ->  th ) ) ) )
3:1,2:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  )  ->  ( ps  ->  ( ph  ->  ( ch  ->  th ) ) ) )
4::  |-  ( ( ps  ->  ( ph  ->  ( ch  ->  th ) ) )  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
5:3,4:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  )  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
6::  |-  ( ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) ) )
qed:5,6:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  )  <->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
 |-  (
 ( ph  ->  ( ps 
 ->  ( ch  ->  ( ph  ->  th ) ) ) )  <->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
 
Theoremee233 28281 Non-virtual deduction form of e233 28540. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
h1::  |-  ( ph  ->  ( ps  ->  ch ) )
h2::  |-  ( ph  ->  ( ps  ->  ( th  ->  ta ) ) )
h3::  |-  ( ph  ->  ( ps  ->  ( th  ->  et ) ) )
h4::  |-  ( ch  ->  ( ta  ->  ( et  ->  ze ) ) )
5:1,4:  |-  ( ph  ->  ( ps  ->  ( ta  ->  ( et  ->  ze ) ) )  )
6:5:  |-  ( ta  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) )  )
7:2,6:  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) ) )
8:7:  |-  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) ) ) )
9:8:  |-  ( th  ->  ( ph  ->  ( ps  ->  ( et  ->  ze ) ) )  )
10:9:  |-  ( ph  ->  ( ps  ->  ( th  ->  ( et  ->  ze ) ) )  )
11:10:  |-  ( et  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) )  )
12:3,11:  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) ) )
13:12:  |-  ( ps  ->  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) ) ) )
14:13:  |-  ( th  ->  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) )  )
qed:14:  |-  ( ph  ->  ( ps  ->  ( th  ->  ze ) ) )
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  ta ) ) )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  et )
 ) )   &    |-  ( ch  ->  ( ta  ->  ( et  ->  ze ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( th  ->  ze )
 ) )
 
Theoremimbi12 28282 Implication form of imbi12i 316. imbi12 28282 is imbi12VD 28649 without virtual deductions and was automatically derived from imbi12VD 28649 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  <->  th )  ->  (
 ( ph  ->  ch )  <->  ( ps  ->  th )
 ) ) )
 
Theoremimbi13 28283 Join three logical equivalences to form equivalence of implications. imbi13 28283 is imbi13VD 28650 without virtual deductions and was automatically derived from imbi13VD 28650 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  <->  th )  ->  (
 ( ta  <->  et )  ->  (
 ( ph  ->  ( ch 
 ->  ta ) )  <->  ( ps  ->  ( th  ->  et )
 ) ) ) ) )
 
Theoremee33 28284 Non-virtual deduction form of e33 28509. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
h1::  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
h2::  |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
h3::  |-  ( th  ->  ( ta  ->  et ) )
4:1,3:  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) ) )
5:4:  |-  ( ta  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
6:2,5:  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )
7:6:  |-  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) )
8:7:  |-  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
qed:8:  |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )
 
Theoremcon5 28285 Bi-conditional contraposition variation. This proof is con5VD 28676 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  -.  ps )  ->  ( -.  ph  ->  ps )
 )
 
Theoremcon5i 28286 Inference form of con5 28285. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph 
 <->  -.  ps )   =>    |-  ( -.  ph  ->  ps )
 
Theoremexlimexi 28287 Inference similar to Theorem 19.23 of [Margaris] p. 90. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  ( E. x ph  ->  ( ph  ->  ps ) )   =>    |-  ( E. x ph 
 ->  ps )
 
Theoremsb5ALT 28288* Equivalence for substitution. Alternate proof of sb5 2039. This proof is sb5ALTVD 28689 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )
 )
 
Theoremeexinst01 28289 exinst01 28397 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  E. x ps   &    |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  A. x ph )   &    |-  ( ch  ->  A. x ch )   =>    |-  ( ph  ->  ch )
 
Theoremeexinst11 28290 exinst11 28398 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  E. x ps )   &    |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  A. x ph )   &    |-  ( ch  ->  A. x ch )   =>    |-  ( ph  ->  ch )
 
Theoremvk15.4j 28291 Excercise 4j of Unit 15 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. This proof is the minimized Hilbert-style axiomatic version of the Fitch-style Natural Deduction proof found on page 442 of Klenk and was automatically derived from that proof. vk15.4j 28291 is vk15.4jVD 28690 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  ( E. x  -.  ph  /\ 
 E. x ( ps 
 /\  -.  ch )
 )   &    |-  ( A. x ch  ->  -.  E. x ( th  /\  ta )
 )   &    |- 
 -.  A. x ( ta 
 ->  ph )   =>    |-  ( -.  E. x  -.  th  ->  -.  A. x ps )
 
Theoremnotnot2ALT 28292 Converse of double negation. Alternate proof of notnot2 104. This proof is notnot2ALTVD 28691 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  -.  ph  ->  ph )
 
Theoremcon3ALT 28293 Contraposition. Alternate proof of con3 126. This proof is con3ALTVD 28692 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( -.  ps  ->  -.  ph ) )
 
Theoremssralv2 28294* Quantification restricted to a subclass for two quantifiers. ssralv 3237 for two quantifiers. The proof of ssralv2 28294 was automatically generated by minimizing the automatically translated proof of ssralv2VD 28642. The automatic translation is by the tools program translatewithout_overwriting.cmd (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  C_  B  /\  C  C_  D )  ->  ( A. x  e.  B  A. y  e.  D  ph  ->  A. x  e.  A  A. y  e.  C  ph ) )
 
Theoremsbc3org 28295 sbcorg 3036 with a 3-disjuncts. This proof is sbc3orgVD 28627 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  V  ->  (
 [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ].
 ch ) ) )
 
Theoremalrim3con13v 28296* Closed form of alrimi 1745 with 2 additional conjuncts having no occurences of the quantifying variable. This proof is 19.21a3con13vVD 28628 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  A. x ph )  ->  ( ( ps  /\  ph  /\  ch )  ->  A. x ( ps 
 /\  ph  /\  ch )
 ) )
 
Theoremrspsbc2 28297* rspsbc 3069 with two quantifying variables. This proof is rspsbc2VD 28631 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  ( C  e.  D  ->  (
 A. x  e.  B  A. y  e.  D  ph  -> 
 [. C  /  y ]. [. A  /  x ].
 ph ) ) )
 
Theoremsbcoreleleq 28298* Substitution of a set variable for another set variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 28635. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  V  ->  (
 [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y
 ) 
 <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A )
 ) )
 
Theoremtratrb 28299* If a class is transitive and any two distinct elements of the class are E-comparable, then every element of that class is transitive. Derived automatically from tratrbVD 28637. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( Tr  A  /\  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) 
 ->  Tr  B )
 
Theorem3ax5 28300 ax-5 1544 for a 3 element left-nested implication. Derived automatically from 3ax5VD 28638. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x ( ph  ->  ( ps  ->  ch )
 )  ->  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) )
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