Theorem sbc2ie 3022

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
sbc2ie.1	|- A e. _V
sbc2ie.2	|- B e. _V
sbc2ie.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
sbc2ie	|- ( [. A / x ]. [. B / y ]. ph <-> ps )

Distinct variable groups:   x, y, A   y, B   ps, x, y

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

Proof of Theorem sbc2ie

Step	Hyp	Ref	Expression
1		sbc2ie.1	. 2 |- A e. _V
2		sbc2ie.2	. 2 |- B e. _V
3		nfv 1516	. . 3 |- F/ x ps
4		nfv 1516	. . 3 |- F/ y ps
5	2	nfth 1452	. . 3 |- F/ x B e. _V
6		sbc2ie.3	. . 3 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
7	3, 4, 5, 6	sbc2iegf 3021	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( [. A / x ]. [. B / y ]. ph <-> ps ) )
8	1, 2, 7	mp2an 423	1 |- ( [. A / x ]. [. B / y ]. ph <-> ps )

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:  sbc3ie  3024