Theorem sbcreug 3070

Description: Interchange class substitution and restricted unique existential 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 2992	. 2 |- ( z = A -> ( [ z / x ] E! y e. B ph <-> [. A / x ]. E! y e. B ph ) )
2		dfsbcq2 2992	. . 3 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	reubidv 2681	. 2 |- ( z = A -> ( E! y e. B [ z / x ] ph <-> E! y e. B [. A / x ]. ph ) )
4		nfcv 2339	. . . 4 |- F/_ x B
5		nfs1v 1958	. . . 4 |- F/ x [ z / x ] ph
6	4, 5	nfreuxy 2672	. . 3 |- F/ x E! y e. B [ z / x ] ph
7		sbequ12 1785	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	7	reubidv 2681	. . 3 |- ( x = z -> ( E! y e. B ph <-> E! y e. B [ z / x ] ph ) )
9	6, 8	sbie 1805	. 2 |- ( [ z / x ] E! y e. B ph <-> E! y e. B [ z / x ] ph )
10	1, 3, 9	vtoclbg 2825	1 |- ( A e. V -> ( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   [wsb 1776   e. wcel 2167   E!wreu 2477   [.wsbc 2989

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-reu 2482  df-v 2765  df-sbc 2990

This theorem is referenced by: (None)