Theorem sbcralg 2939

Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
sbcralg	|- ( A e. V -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )

Distinct variable groups:   y, A   x, B   x, y

Allowed substitution hints:   ph( x, y)   A( x)   B( y)   V( x, y)

Proof of Theorem sbcralg

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2865	. 2 |- ( z = A -> ( [ z / x ] A. y e. B ph <-> [. A / x ]. A. y e. B ph ) )
2		dfsbcq2 2865	. . 3 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	ralbidv 2396	. 2 |- ( z = A -> ( A. y e. B [ z / x ] ph <-> A. y e. B [. A / x ]. ph ) )
4		nfcv 2240	. . . 4 |- F/_ x B
5		nfs1v 1875	. . . 4 |- F/ x [ z / x ] ph
6	4, 5	nfralxy 2430	. . 3 |- F/ x A. y e. B [ z / x ] ph
7		sbequ12 1712	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	7	ralbidv 2396	. . 3 |- ( x = z -> ( A. y e. B ph <-> A. y e. B [ z / x ] ph ) )
9	6, 8	sbie 1732	. 2 |- ( [ z / x ] A. y e. B ph <-> A. y e. B [ z / x ] ph )
10	1, 3, 9	vtoclbg 2702	1 |- ( A e. V -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1299   e. wcel 1448   [wsb 1703   A.wral 2375   [.wsbc 2862

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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-sbc 2863

This theorem is referenced by:  r19.12sn  3536