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Theorem fvsb 27622
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 5454 . . 3  |-  ( F `
 A )  =  ( iota y A F y )
2 dfsbcq 3155 . . 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 27587 . 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 2280   [.wsbc 3153   class class class wbr 4204   iotacio 5408   ` cfv 5446
This theorem is referenced by:  fveqsb  27623
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 2416
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-v 2950  df-sbc 3154  df-un 3317  df-sn 3812  df-pr 3813  df-uni 4008  df-iota 5410  df-fv 5454
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