Theorem sbc2iegf 3068

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

Hypotheses

Ref	Expression
sbc2iegf.1	|- F/ x ps
sbc2iegf.2	|- F/ y ps
sbc2iegf.3	|- F/ x B e. W
sbc2iegf.4	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
sbc2iegf	|- ( ( A e. V /\ B e. W ) -> ( [. A / x ]. [. B / y ]. ph <-> ps ) )

Distinct variable groups:   x, y, A   y, B   x, V   y, W

Allowed substitution hints:   ph( x, y)   ps( x, y)   B( x)   V( y)   W( x)

Proof of Theorem sbc2iegf

Step	Hyp	Ref	Expression
1		simpl 109	. 2 |- ( ( A e. V /\ B e. W ) -> A e. V )
2		simpl 109	. . . 4 |- ( ( B e. W /\ x = A ) -> B e. W )
3		sbc2iegf.4	. . . . 5 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
4	3	adantll 476	. . . 4 |- ( ( ( B e. W /\ x = A ) /\ y = B ) -> ( ph <-> ps ) )
5		nfv 1550	. . . 4 |- F/ y ( B e. W /\ x = A )
6		sbc2iegf.2	. . . . 5 |- F/ y ps
7	6	a1i 9	. . . 4 |- ( ( B e. W /\ x = A ) -> F/ y ps )
8	2, 4, 5, 7	sbciedf 3033	. . 3 |- ( ( B e. W /\ x = A ) -> ( [. B / y ]. ph <-> ps ) )
9	8	adantll 476	. 2 |- ( ( ( A e. V /\ B e. W ) /\ x = A ) -> ( [. B / y ]. ph <-> ps ) )
10		nfv 1550	. . 3 |- F/ x A e. V
11		sbc2iegf.3	. . 3 |- F/ x B e. W
12	10, 11	nfan 1587	. 2 |- F/ x ( A e. V /\ B e. W )
13		sbc2iegf.1	. . 3 |- F/ x ps
14	13	a1i 9	. 2 |- ( ( A e. V /\ B e. W ) -> F/ x ps )
15	1, 9, 12, 14	sbciedf 3033	1 |- ( ( A e. V /\ B e. W ) -> ( [. A / x ]. [. B / y ]. ph <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   F/wnf 1482   e. wcel 2175   [.wsbc 2997

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-sbc 2998

This theorem is referenced by:  sbc2ie  3069  opelopabaf  4319