Theorem sbc2iedv 3071

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 3035	. 2 |- ( ( ph /\ x = A ) -> ( [. B / y ]. ps <-> ch ) )
8	2, 7	sbcied 3035	1 |- ( ph -> ( [. A / x ]. [. B / y ]. ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   _Vcvv 2772   [.wsbc 2998

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sbc 2999

This theorem is referenced by:  dfoprab3  6279  ismnddef  13283