Theorem sbcreug 3017

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 2940	. 2 |- ( z = A -> ( [ z / x ] E! y e. B ph <-> [. A / x ]. E! y e. B ph ) )
2		dfsbcq2 2940	. . 3 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	reubidv 2640	. 2 |- ( z = A -> ( E! y e. B [ z / x ] ph <-> E! y e. B [. A / x ]. ph ) )
4		nfcv 2299	. . . 4 |- F/_ x B
5		nfs1v 1919	. . . 4 |- F/ x [ z / x ] ph
6	4, 5	nfreuxy 2631	. . 3 |- F/ x E! y e. B [ z / x ] ph
7		sbequ12 1751	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	7	reubidv 2640	. . 3 |- ( x = z -> ( E! y e. B ph <-> E! y e. B [ z / x ] ph ) )
9	6, 8	sbie 1771	. 2 |- ( [ z / x ] E! y e. B ph <-> E! y e. B [ z / x ] ph )
10	1, 3, 9	vtoclbg 2773	1 |- ( A e. V -> ( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1335   [wsb 1742   e. wcel 2128   E!wreu 2437   [.wsbc 2937

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-reu 2442  df-v 2714  df-sbc 2938

This theorem is referenced by: (None)