Theorem sbcreug 3045

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 2967	. 2 |- ( z = A -> ( [ z / x ] E! y e. B ph <-> [. A / x ]. E! y e. B ph ) )
2		dfsbcq2 2967	. . 3 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	reubidv 2661	. 2 |- ( z = A -> ( E! y e. B [ z / x ] ph <-> E! y e. B [. A / x ]. ph ) )
4		nfcv 2319	. . . 4 |- F/_ x B
5		nfs1v 1939	. . . 4 |- F/ x [ z / x ] ph
6	4, 5	nfreuxy 2652	. . 3 |- F/ x E! y e. B [ z / x ] ph
7		sbequ12 1771	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	7	reubidv 2661	. . 3 |- ( x = z -> ( E! y e. B ph <-> E! y e. B [ z / x ] ph ) )
9	6, 8	sbie 1791	. 2 |- ( [ z / x ] E! y e. B ph <-> E! y e. B [ z / x ] ph )
10	1, 3, 9	vtoclbg 2800	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 1353   [wsb 1762   e. wcel 2148   E!wreu 2457   [.wsbc 2964

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-reu 2462  df-v 2741  df-sbc 2965

This theorem is referenced by: (None)