Theorem sbcralt 2953

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 2899	. 2 |- ( [. A / z ]. [. z / x ]. A. y e. B ph <-> [. A / x ]. A. y e. B ph )
2		simpl 108	. . 3 |- ( ( A e. V /\ F/_ y A ) -> A e. V )
3		sbsbc 2882	. . . . 5 |- ( [ z / x ] A. y e. B ph <-> [. z / x ]. A. y e. B ph )
4		nfcv 2255	. . . . . . 7 |- F/_ x B
5		nfs1v 1890	. . . . . . 7 |- F/ x [ z / x ] ph
6	4, 5	nfralxy 2445	. . . . . 6 |- F/ x A. y e. B [ z / x ] ph
7		sbequ12 1727	. . . . . . 7 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	7	ralbidv 2411	. . . . . 6 |- ( x = z -> ( A. y e. B ph <-> A. y e. B [ z / x ] ph ) )
9	6, 8	sbie 1747	. . . . 5 |- ( [ z / x ] A. y e. B ph <-> A. y e. B [ z / x ] ph )
10	3, 9	bitr3i 185	. . . 4 |- ( [. z / x ]. A. y e. B ph <-> A. y e. B [ z / x ] ph )
11		nfnfc1 2258	. . . . . . 7 |- F/ y F/_ y A
12		nfcvd 2256	. . . . . . . 8 |- ( F/_ y A -> F/_ y z )
13		id 19	. . . . . . . 8 |- ( F/_ y A -> F/_ y A )
14	12, 13	nfeqd 2270	. . . . . . 7 |- ( F/_ y A -> F/ y z = A )
15	11, 14	nfan1 1526	. . . . . 6 |- F/ y ( F/_ y A /\ z = A )
16		dfsbcq2 2881	. . . . . . 7 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
17	16	adantl 273	. . . . . 6 |- ( ( F/_ y A /\ z = A ) -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
18	15, 17	ralbid 2409	. . . . 5 |- ( ( F/_ y A /\ z = A ) -> ( A. y e. B [ z / x ] ph <-> A. y e. B [. A / x ]. ph ) )
19	18	adantll 465	. . . 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 191	. . 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 2913	. 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 193	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 103   <-> wb 104   = wceq 1314   e. wcel 1463   [wsb 1718   F/_wnfc 2242   A.wral 2390   [.wsbc 2878

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-v 2659  df-sbc 2879

This theorem is referenced by: (None)