Theorem sbc2iedv 3023

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 378	. . 3 |- ( ( ( ph /\ x = A ) /\ y = B ) -> ( ps <-> ch ) )
7	4, 6	sbcied 2987	. 2 |- ( ( ph /\ x = A ) -> ( [. B / y ]. ps <-> ch ) )
8	2, 7	sbcied 2987	1 |- ( ph -> ( [. A / x ]. [. B / y ]. ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   [.wsbc 2951

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952

This theorem is referenced by:  dfoprab3  6159