Theorem sbcralt 3118

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/_yA) -> ( [.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 3068	. 2 |- ( [.A / z ]. [.z / x ].A.y e. B ph <-> [.A / x ].A.y e. B ph)
2		simpl 443	. . 3 |- ((A e. V /\ F/_yA) -> A e. V)
3		sbsbc 3050	. . . . 5 |- ([z / x]A.y e. B ph <-> [.z / x ].A.y e. B ph)
4		nfcv 2489	. . . . . . 7 |- F/_xB
5		nfs1v 2106	. . . . . . 7 |- F/x[z / x]ph
6	4, 5	nfral 2667	. . . . . 6 |- F/xA.y e. B [z / x]ph
7		sbequ12 1919	. . . . . . 7 |- (x = z -> (ph <-> [z / x]ph))
8	7	ralbidv 2634	. . . . . 6 |- (x = z -> (A.y e. B ph <-> A.y e. B [z / x]ph))
9	6, 8	sbie 2038	. . . . 5 |- ([z / x]A.y e. B ph <-> A.y e. B [z / x]ph)
10	3, 9	bitr3i 242	. . . 4 |- ( [.z / x ].A.y e. B ph <-> A.y e. B [z / x]ph)
11		nfnfc1 2492	. . . . . . 7 |- F/y F/_yA
12		nfcvd 2490	. . . . . . . 8 |- ( F/_yA -> F/_yz)
13		id 19	. . . . . . . 8 |- ( F/_yA -> F/_yA)
14	12, 13	nfeqd 2503	. . . . . . 7 |- ( F/_yA -> F/y z = A)
15	11, 14	nfan1 1881	. . . . . 6 |- F/y( F/_yA /\ z = A)
16		dfsbcq2 3049	. . . . . . 7 |- (z = A -> ([z / x]ph <-> [.A / x ].ph))
17	16	adantl 452	. . . . . 6 |- (( F/_yA /\ z = A) -> ([z / x]ph <-> [.A / x ].ph))
18	15, 17	ralbid 2632	. . . . 5 |- (( F/_yA /\ z = A) -> (A.y e. B [z / x]ph <-> A.y e. B [.A / x ].ph))
19	18	adantll 694	. . . 4 |- (((A e. V /\ F/_yA) /\ z = A) -> (A.y e. B [z / x]ph <-> A.y e. B [.A / x ].ph))
20	10, 19	syl5bb 248	. . 3 |- (((A e. V /\ F/_yA) /\ z = A) -> ( [.z / x ].A.y e. B ph <-> A.y e. B [.A / x ].ph))
21	2, 20	sbcied 3082	. 2 |- ((A e. V /\ F/_yA) -> ( [.A / z ]. [.z / x ].A.y e. B ph <-> A.y e. B [.A / x ].ph))
22	1, 21	syl5bbr 250	1 |- ((A e. V /\ F/_yA) -> ( [.A / x ].A.y e. B ph <-> A.y e. B [.A / x ].ph))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642  [wsb 1648   e. wcel 1710   F/_wnfc 2476  A.wral 2614   [.wsbc 3046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-sbc 3047

This theorem is referenced by:  sbcrext  3119