Theorem sbc2ie 2829

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 1421	. . 3 |- F/ x ps
4		nfv 1421	. . 3 |- F/ y ps
5	2	nfth 1353	. . 3 |- F/ x B e. _V
6		sbc2ie.3	. . 3 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
7	3, 4, 5, 6	sbc2iegf 2828	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( [. A / x ]. [. B / y ]. ph <-> ps ) )
8	1, 2, 7	mp2an 402	1 |- ( [. A / x ]. [. B / y ]. ph <-> ps )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   _Vcvv 2557   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765

This theorem is referenced by:  sbc3ie  2831