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Theorem abeq2 2363
Description: Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that abbi 2368 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a class abstraction for each class variable.

Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable  ph (that has a free variable  x) to a theorem with a class variable  A, we substitute  x  e.  A for  ph throughout and simplify, where  A is a new class variable not already in the wff. An example is the conversion of zfauscl 4117 to inex1 4129 (look at the instance of zfauscl 4117 that occurs in the proof of inex1 4129). Conversely, to convert a theorem with a class variable  A to one with 
ph, we substitute  { x  | 
ph } for  A throughout and simplify, where  x and  ph are new set and wff variables not already in the wff. An example is cp 7529, which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 7528. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
abeq2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem abeq2
StepHypRef Expression
1 ax-17 1628 . . 3  |-  ( y  e.  A  ->  A. x  y  e.  A )
2 hbab1 2247 . . 3  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
31, 2cleqh 2355 . 2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  x  e.  { x  | 
ph } ) )
4 abid 2246 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
54bibi2i 306 . . 3  |-  ( ( x  e.  A  <->  x  e.  { x  |  ph }
)  <->  ( x  e.  A  <->  ph ) )
65albii 1554 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  { x  |  ph } )  <->  A. x
( x  e.  A  <->  ph ) )
73, 6bitri 242 1  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   A.wal 1532    = wceq 1619    e. wcel 1621   {cab 2244
This theorem is referenced by:  abeq1  2364  abbi2i  2369  abbi2dv  2373  clabel  2379  sbabel  2420  rabid2  2692  ru  2965  sbcabel  3043  dfss2  3144  zfrep4  4113  pwex  4165  dmopab3  4879  funimaexg  5267
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254
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