Theorem sbc2iegf 3060

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 1542	. . . 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 3025	. . 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 1542	. . 3 |- F/ x A e. V
11		sbc2iegf.3	. . 3 |- F/ x B e. W
12	10, 11	nfan 1579	. 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 3025	1 |- ( ( A e. V /\ B e. W ) -> ( [. A / x ]. [. B / y ]. ph <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   F/wnf 1474   e. wcel 2167   [.wsbc 2989

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990

This theorem is referenced by:  sbc2ie  3061  opelopabaf  4308