Theorem sbc2iegf 3073

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 1552	. . . 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 3038	. . 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 1552	. . 3 |- F/ x A e. V
11		sbc2iegf.3	. . 3 |- F/ x B e. W
12	10, 11	nfan 1589	. 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 3038	1 |- ( ( A e. V /\ B e. W ) -> ( [. A / x ]. [. B / y ]. ph <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   F/wnf 1484   e. wcel 2177   [.wsbc 3002

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3003

This theorem is referenced by:  sbc2ie  3074  opelopabaf  4328  wrd2ind  11199