Theorem sbcralt 2834

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 2785	. 2 |- ( [. A / z ]. [. z / x ]. A. y e. B ph <-> [. A / x ]. A. y e. B ph )
2		simpl 102	. . 3 |- ( ( A e. V /\ F/_ y A ) -> A e. V )
3		sbsbc 2768	. . . . 5 |- ( [ z / x ] A. y e. B ph <-> [. z / x ]. A. y e. B ph )
4		nfcv 2178	. . . . . . 7 |- F/_ x B
5		nfs1v 1815	. . . . . . 7 |- F/ x [ z / x ] ph
6	4, 5	nfralxy 2360	. . . . . 6 |- F/ x A. y e. B [ z / x ] ph
7		sbequ12 1654	. . . . . . 7 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	7	ralbidv 2326	. . . . . 6 |- ( x = z -> ( A. y e. B ph <-> A. y e. B [ z / x ] ph ) )
9	6, 8	sbie 1674	. . . . 5 |- ( [ z / x ] A. y e. B ph <-> A. y e. B [ z / x ] ph )
10	3, 9	bitr3i 175	. . . 4 |- ( [. z / x ]. A. y e. B ph <-> A. y e. B [ z / x ] ph )
11		nfnfc1 2181	. . . . . . 7 |- F/ y F/_ y A
12		nfcvd 2179	. . . . . . . 8 |- ( F/_ y A -> F/_ y z )
13		id 19	. . . . . . . 8 |- ( F/_ y A -> F/_ y A )
14	12, 13	nfeqd 2192	. . . . . . 7 |- ( F/_ y A -> F/ y z = A )
15	11, 14	nfan1 1456	. . . . . 6 |- F/ y ( F/_ y A /\ z = A )
16		dfsbcq2 2767	. . . . . . 7 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
17	16	adantl 262	. . . . . 6 |- ( ( F/_ y A /\ z = A ) -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
18	15, 17	ralbid 2324	. . . . 5 |- ( ( F/_ y A /\ z = A ) -> ( A. y e. B [ z / x ] ph <-> A. y e. B [. A / x ]. ph ) )
19	18	adantll 445	. . . 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 181	. . 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 2799	. 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 183	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   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   [wsb 1645   F/_wnfc 2165   A.wral 2306   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-sbc 2765

This theorem is referenced by: (None)