Theorem sbcralt 3315

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 3265	. 2 |- ( [. A / z ]. [. z / x ]. A. y e. B ph <-> [. A / x ]. A. y e. B ph )
2		simpl 458	. . 3 |- ( ( A e. V /\ F/_ y A ) -> A e. V )
3		sbsbc 3246	. . . . 5 |- ( [ z / x ] A. y e. B ph <-> [. z / x ]. A. y e. B ph )
4		nfcv 2569	. . . . . . 7 |- F/_ x B
5		nfs1v 2243	. . . . . . 7 |- F/ x [ z / x ] ph
6	4, 5	nfral 2751	. . . . . 6 |- F/ x A. y e. B [ z / x ] ph
7		sbequ12 2057	. . . . . . 7 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	7	ralbidv 2804	. . . . . 6 |- ( x = z -> ( A. y e. B ph <-> A. y e. B [ z / x ] ph ) )
9	6, 8	sbie 2213	. . . . 5 |- ( [ z / x ] A. y e. B ph <-> A. y e. B [ z / x ] ph )
10	3, 9	bitr3i 254	. . . 4 |- ( [. z / x ]. A. y e. B ph <-> A. y e. B [ z / x ] ph )
11		nfnfc1 2572	. . . . . . 7 |- F/ y F/_ y A
12		nfcvd 2570	. . . . . . . 8 |- ( F/_ y A -> F/_ y z )
13		id 22	. . . . . . . 8 |- ( F/_ y A -> F/_ y A )
14	12, 13	nfeqd 2576	. . . . . . 7 |- ( F/_ y A -> F/ y z = A )
15	11, 14	nfan1 1987	. . . . . 6 |- F/ y ( F/_ y A /\ z = A )
16		dfsbcq2 3245	. . . . . . 7 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
17	16	adantl 467	. . . . . 6 |- ( ( F/_ y A /\ z = A ) -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
18	15, 17	ralbid 2799	. . . . 5 |- ( ( F/_ y A /\ z = A ) -> ( A. y e. B [ z / x ] ph <-> A. y e. B [. A / x ]. ph ) )
19	18	adantll 718	. . . 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 260	. . 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 3279	. 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 262	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 187   /\ wa 370   = wceq 1437   [wsb 1790   e. wcel 1872   F/_wnfc 2556   A.wral 2714   [.wsbc 3242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-v 3024  df-sbc 3243

This theorem is referenced by:  sbcrext  3316  sbcralg  3317