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Theorem fveqsb 27759
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 5555 . . 3  |-  ( F `
 A )  e. 
_V
2 fveqsb.3 . . . 4  |-  F/ x ps
3 fveqsb.2 . . . 4  |-  ( x  =  ( F `  A )  ->  ( ph 
<->  ps ) )
42, 3sbciegf 3035 . . 3  |-  ( ( F `  A )  e.  _V  ->  ( [. ( F `  A
)  /  x ]. ph  <->  ps ) )
51, 4ax-mp 8 . 2  |-  ( [. ( F `  A )  /  x ]. ph  <->  ps )
6 fvsb 27758 . 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 1530   E.wex 1531   F/wnf 1534    = wceq 1632    e. wcel 1696   E!weu 2156   _Vcvv 2801   [.wsbc 3004   class class class wbr 4039   ` cfv 5271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235  df-fv 5279
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