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Theorem fvsb 27631
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 5462 . . 3  |-  ( F `
 A )  =  ( iota y A F y )
2 dfsbcq 3163 . . 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 27596 . 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 249 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 177    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652   E!weu 2281   [.wsbc 3161   class class class wbr 4212   iotacio 5416   ` cfv 5454
This theorem is referenced by:  fveqsb  27632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-v 2958  df-sbc 3162  df-un 3325  df-sn 3820  df-pr 3821  df-uni 4016  df-iota 5418  df-fv 5462
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