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Theorem setindtrs 27221
Description: Epsilon induction scheme without Infinity. See comments at setindtr 27220. (Contributed by Stefan O'Rear, 28-Oct-2014.)
Hypotheses
Ref Expression
setindtrs.a  |-  ( A. y  e.  x  ps  ->  ph )
setindtrs.b  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
setindtrs.c  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
Assertion
Ref Expression
setindtrs  |-  ( E. z ( Tr  z  /\  B  e.  z
)  ->  ch )
Distinct variable groups:    x, B, z    ph, y    ps, x    ch, x    ph, z    x, y
Allowed substitution hints:    ph( x)    ps( y, z)    ch( y, z)    B( y)

Proof of Theorem setindtrs
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 setindtr 27220 . . 3  |-  ( A. a ( a  C_  { x  |  ph }  ->  a  e.  { x  |  ph } )  -> 
( E. z ( Tr  z  /\  B  e.  z )  ->  B  e.  { x  |  ph } ) )
2 dfss3 3183 . . . 4  |-  ( a 
C_  { x  | 
ph }  <->  A. y  e.  a  y  e.  { x  |  ph }
)
3 nfcv 2432 . . . . . . 7  |-  F/_ x
a
4 nfsab1 2286 . . . . . . 7  |-  F/ x  y  e.  { x  |  ph }
53, 4nfral 2609 . . . . . 6  |-  F/ x A. y  e.  a 
y  e.  { x  |  ph }
6 nfsab1 2286 . . . . . 6  |-  F/ x  a  e.  { x  |  ph }
75, 6nfim 1781 . . . . 5  |-  F/ x
( A. y  e.  a  y  e.  {
x  |  ph }  ->  a  e.  { x  |  ph } )
8 raleq 2749 . . . . . 6  |-  ( x  =  a  ->  ( A. y  e.  x  y  e.  { x  |  ph }  <->  A. y  e.  a  y  e.  { x  |  ph }
) )
9 eleq1 2356 . . . . . 6  |-  ( x  =  a  ->  (
x  e.  { x  |  ph }  <->  a  e.  { x  |  ph }
) )
108, 9imbi12d 311 . . . . 5  |-  ( x  =  a  ->  (
( A. y  e.  x  y  e.  {
x  |  ph }  ->  x  e.  { x  |  ph } )  <->  ( A. y  e.  a  y  e.  { x  |  ph }  ->  a  e.  {
x  |  ph }
) ) )
11 setindtrs.a . . . . . 6  |-  ( A. y  e.  x  ps  ->  ph )
12 vex 2804 . . . . . . . 8  |-  y  e. 
_V
13 setindtrs.b . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1412, 13elab 2927 . . . . . . 7  |-  ( y  e.  { x  | 
ph }  <->  ps )
1514ralbii 2580 . . . . . 6  |-  ( A. y  e.  x  y  e.  { x  |  ph } 
<-> 
A. y  e.  x  ps )
16 abid 2284 . . . . . 6  |-  ( x  e.  { x  | 
ph }  <->  ph )
1711, 15, 163imtr4i 257 . . . . 5  |-  ( A. y  e.  x  y  e.  { x  |  ph }  ->  x  e.  {
x  |  ph }
)
187, 10, 17chvar 1939 . . . 4  |-  ( A. y  e.  a  y  e.  { x  |  ph }  ->  a  e.  {
x  |  ph }
)
192, 18sylbi 187 . . 3  |-  ( a 
C_  { x  | 
ph }  ->  a  e.  { x  |  ph } )
201, 19mpg 1538 . 2  |-  ( E. z ( Tr  z  /\  B  e.  z
)  ->  B  e.  { x  |  ph }
)
21 elex 2809 . . . . 5  |-  ( B  e.  z  ->  B  e.  _V )
2221adantl 452 . . . 4  |-  ( ( Tr  z  /\  B  e.  z )  ->  B  e.  _V )
2322exlimiv 1624 . . 3  |-  ( E. z ( Tr  z  /\  B  e.  z
)  ->  B  e.  _V )
24 setindtrs.c . . . 4  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
2524elabg 2928 . . 3  |-  ( B  e.  _V  ->  ( B  e.  { x  |  ph }  <->  ch )
)
2623, 25syl 15 . 2  |-  ( E. z ( Tr  z  /\  B  e.  z
)  ->  ( B  e.  { x  |  ph } 
<->  ch ) )
2720, 26mpbid 201 1  |-  ( E. z ( Tr  z  /\  B  e.  z
)  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   _Vcvv 2801    C_ wss 3165   Tr wtr 4129
This theorem is referenced by:  dford3lem2  27223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-reg 7322
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-uni 3844  df-tr 4130
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