Theorem sbc2iedv 3104

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Hypotheses

Ref	Expression
sbc2iedv.1	|- A e. _V
sbc2iedv.2	|- B e. _V
sbc2iedv.3	|- ( ph -> ( ( x = A /\ y = B ) -> ( ps <-> ch ) ) )

Assertion

Ref	Expression
sbc2iedv	|- ( ph -> ( [. A / x ]. [. B / y ]. ps <-> ch ) )

Distinct variable groups:   x, y, A   y, B   ph, x, y   ch, x, y

Allowed substitution hints:   ps( x, y)   B( x)

Proof of Theorem sbc2iedv

Step	Hyp	Ref	Expression
1		sbc2iedv.1	. . 3 |- A e. _V
2	1	a1i 9	. 2 |- ( ph -> A e. _V )
3		sbc2iedv.2	. . . 4 |- B e. _V
4	3	a1i 9	. . 3 |- ( ( ph /\ x = A ) -> B e. _V )
5		sbc2iedv.3	. . . 4 |- ( ph -> ( ( x = A /\ y = B ) -> ( ps <-> ch ) ) )
6	5	impl 380	. . 3 |- ( ( ( ph /\ x = A ) /\ y = B ) -> ( ps <-> ch ) )
7	4, 6	sbcied 3068	. 2 |- ( ( ph /\ x = A ) -> ( [. B / y ]. ps <-> ch ) )
8	2, 7	sbcied 3068	1 |- ( ph -> ( [. A / x ]. [. B / y ]. ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   [.wsbc 3031

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sbc 3032

This theorem is referenced by:  dfoprab3  6353  ismnddef  13500