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Theorem setindtrs 29327
Description: Epsilon induction scheme without Infinity. See comments at setindtr 29326. (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 29326 . . 3  |-  ( A. a ( a  C_  { x  |  ph }  ->  a  e.  { x  |  ph } )  -> 
( E. z ( Tr  z  /\  B  e.  z )  ->  B  e.  { x  |  ph } ) )
2 dfss3 3341 . . . 4  |-  ( a 
C_  { x  | 
ph }  <->  A. y  e.  a  y  e.  { x  |  ph }
)
3 nfcv 2574 . . . . . . 7  |-  F/_ x
a
4 nfsab1 2428 . . . . . . 7  |-  F/ x  y  e.  { x  |  ph }
53, 4nfral 2764 . . . . . 6  |-  F/ x A. y  e.  a 
y  e.  { x  |  ph }
6 nfsab1 2428 . . . . . 6  |-  F/ x  a  e.  { x  |  ph }
75, 6nfim 1852 . . . . 5  |-  F/ x
( A. y  e.  a  y  e.  {
x  |  ph }  ->  a  e.  { x  |  ph } )
8 raleq 2912 . . . . . 6  |-  ( x  =  a  ->  ( A. y  e.  x  y  e.  { x  |  ph }  <->  A. y  e.  a  y  e.  { x  |  ph }
) )
9 eleq1 2498 . . . . . 6  |-  ( x  =  a  ->  (
x  e.  { x  |  ph }  <->  a  e.  { x  |  ph }
) )
108, 9imbi12d 320 . . . . 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 2970 . . . . . . . 8  |-  y  e. 
_V
13 setindtrs.b . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1412, 13elab 3101 . . . . . . 7  |-  ( y  e.  { x  | 
ph }  <->  ps )
1514ralbii 2734 . . . . . 6  |-  ( A. y  e.  x  y  e.  { x  |  ph } 
<-> 
A. y  e.  x  ps )
16 abid 2426 . . . . . 6  |-  ( x  e.  { x  | 
ph }  <->  ph )
1711, 15, 163imtr4i 266 . . . . 5  |-  ( A. y  e.  x  y  e.  { x  |  ph }  ->  x  e.  {
x  |  ph }
)
187, 10, 17chvar 1957 . . . 4  |-  ( A. y  e.  a  y  e.  { x  |  ph }  ->  a  e.  {
x  |  ph }
)
192, 18sylbi 195 . . 3  |-  ( a 
C_  { x  | 
ph }  ->  a  e.  { x  |  ph } )
201, 19mpg 1593 . 2  |-  ( E. z ( Tr  z  /\  B  e.  z
)  ->  B  e.  { x  |  ph }
)
21 elex 2976 . . . . 5  |-  ( B  e.  z  ->  B  e.  _V )
2221adantl 466 . . . 4  |-  ( ( Tr  z  /\  B  e.  z )  ->  B  e.  _V )
2322exlimiv 1688 . . 3  |-  ( E. z ( Tr  z  /\  B  e.  z
)  ->  B  e.  _V )
24 setindtrs.c . . . 4  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
2524elabg 3102 . . 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 210 1  |-  ( E. z ( Tr  z  /\  B  e.  z
)  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2424   A.wral 2710   _Vcvv 2967    C_ wss 3323   Tr wtr 4380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-reg 7799
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-v 2969  df-dif 3326  df-in 3330  df-ss 3337  df-nul 3633  df-uni 4087  df-tr 4381
This theorem is referenced by:  dford3lem2  29329
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