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Theorem setindtrs 26788
Description: Epsilon induction scheme without Infinity. See comments at setindtr 26787. (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 26787 . . 3  |-  ( A. a ( a  C_  { x  |  ph }  ->  a  e.  { x  |  ph } )  -> 
( E. z ( Tr  z  /\  B  e.  z )  ->  B  e.  { x  |  ph } ) )
2 dfss3 3282 . . . 4  |-  ( a 
C_  { x  | 
ph }  <->  A. y  e.  a  y  e.  { x  |  ph }
)
3 nfcv 2524 . . . . . . 7  |-  F/_ x
a
4 nfsab1 2378 . . . . . . 7  |-  F/ x  y  e.  { x  |  ph }
53, 4nfral 2703 . . . . . 6  |-  F/ x A. y  e.  a 
y  e.  { x  |  ph }
6 nfsab1 2378 . . . . . 6  |-  F/ x  a  e.  { x  |  ph }
75, 6nfim 1822 . . . . 5  |-  F/ x
( A. y  e.  a  y  e.  {
x  |  ph }  ->  a  e.  { x  |  ph } )
8 raleq 2848 . . . . . 6  |-  ( x  =  a  ->  ( A. y  e.  x  y  e.  { x  |  ph }  <->  A. y  e.  a  y  e.  { x  |  ph }
) )
9 eleq1 2448 . . . . . 6  |-  ( x  =  a  ->  (
x  e.  { x  |  ph }  <->  a  e.  { x  |  ph }
) )
108, 9imbi12d 312 . . . . 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 2903 . . . . . . . 8  |-  y  e. 
_V
13 setindtrs.b . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1412, 13elab 3026 . . . . . . 7  |-  ( y  e.  { x  | 
ph }  <->  ps )
1514ralbii 2674 . . . . . 6  |-  ( A. y  e.  x  y  e.  { x  |  ph } 
<-> 
A. y  e.  x  ps )
16 abid 2376 . . . . . 6  |-  ( x  e.  { x  | 
ph }  <->  ph )
1711, 15, 163imtr4i 258 . . . . 5  |-  ( A. y  e.  x  y  e.  { x  |  ph }  ->  x  e.  {
x  |  ph }
)
187, 10, 17chvar 2023 . . . 4  |-  ( A. y  e.  a  y  e.  { x  |  ph }  ->  a  e.  {
x  |  ph }
)
192, 18sylbi 188 . . 3  |-  ( a 
C_  { x  | 
ph }  ->  a  e.  { x  |  ph } )
201, 19mpg 1554 . 2  |-  ( E. z ( Tr  z  /\  B  e.  z
)  ->  B  e.  { x  |  ph }
)
21 elex 2908 . . . . 5  |-  ( B  e.  z  ->  B  e.  _V )
2221adantl 453 . . . 4  |-  ( ( Tr  z  /\  B  e.  z )  ->  B  e.  _V )
2322exlimiv 1641 . . 3  |-  ( E. z ( Tr  z  /\  B  e.  z
)  ->  B  e.  _V )
24 setindtrs.c . . . 4  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
2524elabg 3027 . . 3  |-  ( B  e.  _V  ->  ( B  e.  { x  |  ph }  <->  ch )
)
2623, 25syl 16 . 2  |-  ( E. z ( Tr  z  /\  B  e.  z
)  ->  ( B  e.  { x  |  ph } 
<->  ch ) )
2720, 26mpbid 202 1  |-  ( E. z ( Tr  z  /\  B  e.  z
)  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2374   A.wral 2650   _Vcvv 2900    C_ wss 3264   Tr wtr 4244
This theorem is referenced by:  dford3lem2  26790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-reg 7494
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-v 2902  df-dif 3267  df-in 3271  df-ss 3278  df-nul 3573  df-uni 3959  df-tr 4245
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