Theorem fvsb 38656

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

Step	Hyp	Ref	Expression
1		df-fv 5896	. . 3 |- ( F ` A ) = ( iota y A F y )
2		dfsbcq 3437	. . 3 |- ( ( F ` A ) = ( iota y A F y ) -> ( [. ( F ` A ) / x ]. ph <-> [. ( iota y A F y ) / x ]. ph ) )
3	1, 2	ax-mp 5	. 2 |- ( [. ( F ` A ) / x ]. ph <-> [. ( iota y A F y ) / x ]. ph )
4		iotasbc 38620	. 2 |- ( E! y A F y -> ( [. ( iota y A F y ) / x ]. ph <-> E. x ( A. y ( A F y <-> y = x ) /\ ph ) ) )
5	3, 4	syl5bb 272	1 |- ( E! y A F y -> ( [. ( F ` A ) / x ]. ph <-> E. x ( A. y ( A F y <-> y = x ) /\ ph ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   E!weu 2470   [.wsbc 3435   class class class wbr 4653   iotacio 5849   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851  df-fv 5896

This theorem is referenced by:  fveqsb  38657