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Theorem fvsb 26978
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 5345 . . 3  |-  ( F `
 A )  =  ( iota y A F y )
2 dfsbcq 3069 . . 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 26942 . 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 1540   E.wex 1541    = wceq 1642   E!weu 2209   [.wsbc 3067   class class class wbr 4104   iotacio 5299   ` cfv 5337
This theorem is referenced by:  fveqsb  26979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rex 2625  df-v 2866  df-sbc 3068  df-un 3233  df-sn 3722  df-pr 3723  df-uni 3909  df-iota 5301  df-fv 5345
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