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Theorem setindtrs 29515
Description: Epsilon induction scheme without Infinity. See comments at setindtr 29514. (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 29514 . . 3  |-  ( A. a ( a  C_  { x  |  ph }  ->  a  e.  { x  |  ph } )  -> 
( E. z ( Tr  z  /\  B  e.  z )  ->  B  e.  { x  |  ph } ) )
2 dfss3 3447 . . . 4  |-  ( a 
C_  { x  | 
ph }  <->  A. y  e.  a  y  e.  { x  |  ph }
)
3 nfcv 2613 . . . . . . 7  |-  F/_ x
a
4 nfsab1 2440 . . . . . . 7  |-  F/ x  y  e.  { x  |  ph }
53, 4nfral 2881 . . . . . 6  |-  F/ x A. y  e.  a 
y  e.  { x  |  ph }
6 nfsab1 2440 . . . . . 6  |-  F/ x  a  e.  { x  |  ph }
75, 6nfim 1855 . . . . 5  |-  F/ x
( A. y  e.  a  y  e.  {
x  |  ph }  ->  a  e.  { x  |  ph } )
8 raleq 3016 . . . . . 6  |-  ( x  =  a  ->  ( A. y  e.  x  y  e.  { x  |  ph }  <->  A. y  e.  a  y  e.  { x  |  ph }
) )
9 eleq1 2523 . . . . . 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 3074 . . . . . . . 8  |-  y  e. 
_V
13 setindtrs.b . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1412, 13elab 3206 . . . . . . 7  |-  ( y  e.  { x  | 
ph }  <->  ps )
1514ralbii 2834 . . . . . 6  |-  ( A. y  e.  x  y  e.  { x  |  ph } 
<-> 
A. y  e.  x  ps )
16 abid 2438 . . . . . 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 1966 . . . 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 1594 . 2  |-  ( E. z ( Tr  z  /\  B  e.  z
)  ->  B  e.  { x  |  ph }
)
21 elex 3080 . . . . 5  |-  ( B  e.  z  ->  B  e.  _V )
2221adantl 466 . . . 4  |-  ( ( Tr  z  /\  B  e.  z )  ->  B  e.  _V )
2322exlimiv 1689 . . 3  |-  ( E. z ( Tr  z  /\  B  e.  z
)  ->  B  e.  _V )
24 setindtrs.c . . . 4  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
2524elabg 3207 . . 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 1370   E.wex 1587    e. wcel 1758   {cab 2436   A.wral 2795   _Vcvv 3071    C_ wss 3429   Tr wtr 4486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-reg 7911
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-v 3073  df-dif 3432  df-in 3436  df-ss 3443  df-nul 3739  df-uni 4193  df-tr 4487
This theorem is referenced by:  dford3lem2  29517
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