Theorem sbcralg 3029

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 2954	. 2 |- ( z = A -> ( [ z / x ] A. y e. B ph <-> [. A / x ]. A. y e. B ph ) )
2		dfsbcq2 2954	. . 3 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	ralbidv 2466	. 2 |- ( z = A -> ( A. y e. B [ z / x ] ph <-> A. y e. B [. A / x ]. ph ) )
4		nfcv 2308	. . . 4 |- F/_ x B
5		nfs1v 1927	. . . 4 |- F/ x [ z / x ] ph
6	4, 5	nfralxy 2504	. . 3 |- F/ x A. y e. B [ z / x ] ph
7		sbequ12 1759	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	7	ralbidv 2466	. . 3 |- ( x = z -> ( A. y e. B ph <-> A. y e. B [ z / x ] ph ) )
9	6, 8	sbie 1779	. 2 |- ( [ z / x ] A. y e. B ph <-> A. y e. B [ z / x ] ph )
10	1, 3, 9	vtoclbg 2787	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 1343   [wsb 1750   e. wcel 2136   A.wral 2444   [.wsbc 2951

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-sbc 2952

This theorem is referenced by:  r19.12sn  3642