Theorem fveqsb 38657

Description: Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)

Hypotheses

Ref	Expression
fveqsb.2	|- ( x = ( F ` A ) -> ( ph <-> ps ) )
fveqsb.3	|- F/ x ps

Assertion

Ref	Expression
fveqsb	|- ( E! y A F y -> ( ps <-> 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)   ps( x, y)

Proof of Theorem fveqsb

Step	Hyp	Ref	Expression
1		fvex 6201	. . 3 |- ( F ` A ) e. _V
2		fveqsb.3	. . . 4 |- F/ x ps
3		fveqsb.2	. . . 4 |- ( x = ( F ` A ) -> ( ph <-> ps ) )
4	2, 3	sbciegf 3467	. . 3 |- ( ( F ` A ) e. _V -> ( [. ( F ` A ) / x ]. ph <-> ps ) )
5	1, 4	ax-mp 5	. 2 |- ( [. ( F ` A ) / x ]. ph <-> ps )
6		fvsb 38656	. 2 |- ( E! y A F y -> ( [. ( F ` A ) / x ]. ph <-> E. x ( A. y ( A F y <-> y = x ) /\ ph ) ) )
7	5, 6	syl5bbr 274	1 |- ( E! y A F y -> ( ps <-> 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   F/wnf 1708   e. wcel 1990   E!weu 2470   _Vcvv 3200   [.wsbc 3435   class class class wbr 4653   ` 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  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851  df-fv 5896

This theorem is referenced by: (None)