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Theorem fvsb 27758
Description: Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
Assertion
Ref Expression
fvsb  |-  ( E! y  A F y  ->  ( [. ( F `  A )  /  x ]. ph  <->  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)

Proof of Theorem fvsb
StepHypRef Expression
1 df-fv 5279 . . 3  |-  ( F `
 A )  =  ( iota y A F y )
2 dfsbcq 3006 . . 3  |-  ( ( F `  A )  =  ( iota y A F y )  -> 
( [. ( F `  A )  /  x ]. ph  <->  [. ( iota y A F y )  /  x ]. ph ) )
31, 2ax-mp 8 . 2  |-  ( [. ( F `  A )  /  x ]. ph  <->  [. ( iota y A F y )  /  x ]. ph )
4 iotasbc 27722 . 2  |-  ( E! y  A F y  ->  ( [. ( iota y A F y )  /  x ]. ph  <->  E. x ( A. y
( A F y  <-> 
y  =  x )  /\  ph ) ) )
53, 4syl5bb 248 1  |-  ( E! y  A F y  ->  ( [. ( F `  A )  /  x ]. ph  <->  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    = wceq 1632   E!weu 2156   [.wsbc 3004   class class class wbr 4039   iotacio 5233   ` cfv 5271
This theorem is referenced by:  fveqsb  27759
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rex 2562  df-v 2803  df-sbc 3005  df-un 3170  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235  df-fv 5279
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