Theorem sbcreug 2894

Description: Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)

Assertion

Ref	Expression
sbcreug	|- ( A e. V -> ( [. A / x ]. E! y e. B ph <-> E! 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 sbcreug

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2818	. 2 |- ( z = A -> ( [ z / x ] E! y e. B ph <-> [. A / x ]. E! y e. B ph ) )
2		dfsbcq2 2818	. . 3 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	reubidv 2537	. 2 |- ( z = A -> ( E! y e. B [ z / x ] ph <-> E! y e. B [. A / x ]. ph ) )
4		nfcv 2219	. . . 4 |- F/_ x B
5		nfs1v 1856	. . . 4 |- F/ x [ z / x ] ph
6	4, 5	nfreuxy 2528	. . 3 |- F/ x E! y e. B [ z / x ] ph
7		sbequ12 1694	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	7	reubidv 2537	. . 3 |- ( x = z -> ( E! y e. B ph <-> E! y e. B [ z / x ] ph ) )
9	6, 8	sbie 1714	. 2 |- ( [ z / x ] E! y e. B ph <-> E! y e. B [ z / x ] ph )
10	1, 3, 9	vtoclbg 2659	1 |- ( A e. V -> ( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   [wsb 1685   E!wreu 2350   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-reu 2355  df-v 2603  df-sbc 2816

This theorem is referenced by: (None)