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Theorem fveqsb 27317
Description: Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
Hypotheses
Ref Expression
fveqsb.2  |-  ( x  =  ( F `  A )  ->  ( ph 
<->  ps ) )
fveqsb.3  |-  F/ x ps
Assertion
Ref Expression
fveqsb  |-  ( E! y  A F y  ->  ( ps  <->  E. x
( A. y ( A F y  <->  y  =  x )  /\  ph ) ) )
Distinct variable groups:    x, A, y    x, F, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem fveqsb
StepHypRef Expression
1 fvex 5675 . . 3  |-  ( F `
 A )  e. 
_V
2 fveqsb.3 . . . 4  |-  F/ x ps
3 fveqsb.2 . . . 4  |-  ( x  =  ( F `  A )  ->  ( ph 
<->  ps ) )
42, 3sbciegf 3128 . . 3  |-  ( ( F `  A )  e.  _V  ->  ( [. ( F `  A
)  /  x ]. ph  <->  ps ) )
51, 4ax-mp 8 . 2  |-  ( [. ( F `  A )  /  x ]. ph  <->  ps )
6 fvsb 27316 . 2  |-  ( E! y  A F y  ->  ( [. ( F `  A )  /  x ]. ph  <->  E. x
( A. y ( A F y  <->  y  =  x )  /\  ph ) ) )
75, 6syl5bbr 251 1  |-  ( E! y  A F y  ->  ( ps  <->  E. x
( A. y ( A F y  <->  y  =  x )  /\  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547   F/wnf 1550    = wceq 1649    e. wcel 1717   E!weu 2231   _Vcvv 2892   [.wsbc 3097   class class class wbr 4146   ` cfv 5387
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 2361  ax-nul 4272
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-sn 3756  df-pr 3757  df-uni 3951  df-iota 5351  df-fv 5395
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