Theorem sbcralt 3351

Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)

Assertion

Ref	Expression
sbcralt	|- ( ( A e. V /\ F/_ y A ) -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )

Distinct variable groups:   x, y   x, B

Allowed substitution hints:   ph( x, y)   A( x, y)   B( y)   V( x, y)

Proof of Theorem sbcralt

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcco 3301	. 2 |- ( [. A / z ]. [. z / x ]. A. y e. B ph <-> [. A / x ]. A. y e. B ph )
2		simpl 463	. . 3 |- ( ( A e. V /\ F/_ y A ) -> A e. V )
3		sbsbc 3282	. . . . 5 |- ( [ z / x ] A. y e. B ph <-> [. z / x ]. A. y e. B ph )
4		nfcv 2602	. . . . . . 7 |- F/_ x B
5		nfs1v 2276	. . . . . . 7 |- F/ x [ z / x ] ph
6	4, 5	nfral 2785	. . . . . 6 |- F/ x A. y e. B [ z / x ] ph
7		sbequ12 2093	. . . . . . 7 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	7	ralbidv 2838	. . . . . 6 |- ( x = z -> ( A. y e. B ph <-> A. y e. B [ z / x ] ph ) )
9	6, 8	sbie 2247	. . . . 5 |- ( [ z / x ] A. y e. B ph <-> A. y e. B [ z / x ] ph )
10	3, 9	bitr3i 259	. . . 4 |- ( [. z / x ]. A. y e. B ph <-> A. y e. B [ z / x ] ph )
11		nfnfc1 2605	. . . . . . 7 |- F/ y F/_ y A
12		nfcvd 2603	. . . . . . . 8 |- ( F/_ y A -> F/_ y z )
13		id 22	. . . . . . . 8 |- ( F/_ y A -> F/_ y A )
14	12, 13	nfeqd 2609	. . . . . . 7 |- ( F/_ y A -> F/ y z = A )
15	11, 14	nfan1 2020	. . . . . 6 |- F/ y ( F/_ y A /\ z = A )
16		dfsbcq2 3281	. . . . . . 7 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
17	16	adantl 472	. . . . . 6 |- ( ( F/_ y A /\ z = A ) -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
18	15, 17	ralbid 2833	. . . . 5 |- ( ( F/_ y A /\ z = A ) -> ( A. y e. B [ z / x ] ph <-> A. y e. B [. A / x ]. ph ) )
19	18	adantll 725	. . . 4 |- ( ( ( A e. V /\ F/_ y A ) /\ z = A ) -> ( A. y e. B [ z / x ] ph <-> A. y e. B [. A / x ]. ph ) )
20	10, 19	syl5bb 265	. . 3 |- ( ( ( A e. V /\ F/_ y A ) /\ z = A ) -> ( [. z / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )
21	2, 20	sbcied 3315	. 2 |- ( ( A e. V /\ F/_ y A ) -> ( [. A / z ]. [. z / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )
22	1, 21	syl5bbr 267	1 |- ( ( A e. V /\ F/_ y A ) -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1454   [wsb 1807   e. wcel 1897   F/_wnfc 2589   A.wral 2748   [.wsbc 3278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ral 2753  df-v 3058  df-sbc 3279

This theorem is referenced by:  sbcrext  3352  sbcralg  3353