Theorem sbcralg 3076

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 3000	. 2 |- ( z = A -> ( [ z / x ] A. y e. B ph <-> [. A / x ]. A. y e. B ph ) )
2		dfsbcq2 3000	. . 3 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	ralbidv 2505	. 2 |- ( z = A -> ( A. y e. B [ z / x ] ph <-> A. y e. B [. A / x ]. ph ) )
4		nfcv 2347	. . . 4 |- F/_ x B
5		nfs1v 1966	. . . 4 |- F/ x [ z / x ] ph
6	4, 5	nfralxy 2543	. . 3 |- F/ x A. y e. B [ z / x ] ph
7		sbequ12 1793	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	7	ralbidv 2505	. . 3 |- ( x = z -> ( A. y e. B ph <-> A. y e. B [ z / x ] ph ) )
9	6, 8	sbie 1813	. 2 |- ( [ z / x ] A. y e. B ph <-> A. y e. B [ z / x ] ph )
10	1, 3, 9	vtoclbg 2833	1 |- ( A e. V -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1372   [wsb 1784   e. wcel 2175   A.wral 2483   [.wsbc 2997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-sbc 2998

This theorem is referenced by:  r19.12sn  3698