Theorem sbcreug 2993

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 2916	. 2 |- ( z = A -> ( [ z / x ] E! y e. B ph <-> [. A / x ]. E! y e. B ph ) )
2		dfsbcq2 2916	. . 3 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	reubidv 2617	. 2 |- ( z = A -> ( E! y e. B [ z / x ] ph <-> E! y e. B [. A / x ]. ph ) )
4		nfcv 2282	. . . 4 |- F/_ x B
5		nfs1v 1913	. . . 4 |- F/ x [ z / x ] ph
6	4, 5	nfreuxy 2608	. . 3 |- F/ x E! y e. B [ z / x ] ph
7		sbequ12 1745	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	7	reubidv 2617	. . 3 |- ( x = z -> ( E! y e. B ph <-> E! y e. B [ z / x ] ph ) )
9	6, 8	sbie 1765	. 2 |- ( [ z / x ] E! y e. B ph <-> E! y e. B [ z / x ] ph )
10	1, 3, 9	vtoclbg 2750	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 1332   e. wcel 1481   [wsb 1736   E!wreu 2419   [.wsbc 2913

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-reu 2424  df-v 2691  df-sbc 2914

This theorem is referenced by: (None)