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Theorem fveqsb 27068
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 5539 . . 3  |-  ( F `
 A )  e. 
_V
2 fveqsb.3 . . . 4  |-  F/ x ps
3 fveqsb.2 . . . 4  |-  ( x  =  ( F `  A )  ->  ( ph 
<->  ps ) )
42, 3sbciegf 3022 . . 3  |-  ( ( F `  A )  e.  _V  ->  ( [. ( F `  A
)  /  x ]. ph  <->  ps ) )
51, 4ax-mp 8 . 2  |-  ( [. ( F `  A )  /  x ]. ph  <->  ps )
6 fvsb 27067 . 2  |-  ( E! y  A F y  ->  ( [. ( F `  A )  /  x ]. ph  <->  E. x
( A. y ( A F y  <->  y  =  x )  /\  ph ) ) )
75, 6syl5bbr 250 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 176    /\ wa 358   A.wal 1527   E.wex 1528   F/wnf 1531    = wceq 1623    e. wcel 1684   E!weu 2143   _Vcvv 2788   [.wsbc 2991   class class class wbr 4023   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219  df-fv 5263
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