Theorem sbcralt 3031

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 2976	. 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 2959	. . . . 5 |- ( [ z / x ] A. y e. B ph <-> [. z / x ]. A. y e. B ph )
4		nfcv 2312	. . . . . . 7 |- F/_ x B
5		nfs1v 1932	. . . . . . 7 |- F/ x [ z / x ] ph
6	4, 5	nfralxy 2508	. . . . . 6 |- F/ x A. y e. B [ z / x ] ph
7		sbequ12 1764	. . . . . . 7 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	7	ralbidv 2470	. . . . . 6 |- ( x = z -> ( A. y e. B ph <-> A. y e. B [ z / x ] ph ) )
9	6, 8	sbie 1784	. . . . 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 2315	. . . . . . 7 |- F/ y F/_ y A
12		nfcvd 2313	. . . . . . . 8 |- ( F/_ y A -> F/_ y z )
13		id 19	. . . . . . . 8 |- ( F/_ y A -> F/_ y A )
14	12, 13	nfeqd 2327	. . . . . . 7 |- ( F/_ y A -> F/ y z = A )
15	11, 14	nfan1 1557	. . . . . 6 |- F/ y ( F/_ y A /\ z = A )
16		dfsbcq2 2958	. . . . . . 7 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
17	16	adantl 275	. . . . . 6 |- ( ( F/_ y A /\ z = A ) -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
18	15, 17	ralbid 2468	. . . . 5 |- ( ( F/_ y A /\ z = A ) -> ( A. y e. B [ z / x ] ph <-> A. y e. B [. A / x ]. ph ) )
19	18	adantll 473	. . . 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 2991	. 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	bitr3id 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 1348   [wsb 1755   e. wcel 2141   F/_wnfc 2299   A.wral 2448   [.wsbc 2955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-sbc 2956

This theorem is referenced by: (None)