Theorem sbc2iegf 3113

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

Hypotheses

Ref	Expression
sbc2iegf.1	|- F/xps
sbc2iegf.2	|- F/yps
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 443	. 2 |- ((A e. V /\ B e. W) -> A e. V)
2		simpl 443	. . . 4 |- ((B e. W /\ x = A) -> B e. W)
3		sbc2iegf.4	. . . . 5 |- ((x = A /\ y = B) -> (ph <-> ps))
4	3	adantll 694	. . . 4 |- (((B e. W /\ x = A) /\ y = B) -> (ph <-> ps))
5		nfv 1619	. . . 4 |- F/y(B e. W /\ x = A)
6		sbc2iegf.2	. . . . 5 |- F/yps
7	6	a1i 10	. . . 4 |- ((B e. W /\ x = A) -> F/yps)
8	2, 4, 5, 7	sbciedf 3082	. . 3 |- ((B e. W /\ x = A) -> ( [.B / y ].ph <-> ps))
9	8	adantll 694	. 2 |- (((A e. V /\ B e. W) /\ x = A) -> ( [.B / y ].ph <-> ps))
10		nfv 1619	. . 3 |- F/x A e. V
11		sbc2iegf.3	. . 3 |- F/x B e. W
12	10, 11	nfan 1824	. 2 |- F/x(A e. V /\ B e. W)
13		sbc2iegf.1	. . 3 |- F/xps
14	13	a1i 10	. 2 |- ((A e. V /\ B e. W) -> F/xps)
15	1, 9, 12, 14	sbciedf 3082	1 |- ((A e. V /\ B e. W) -> ( [.A / x ]. [.B / y ].ph <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   F/wnf 1544   = wceq 1642   e. wcel 1710   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by:  sbc2ie  3114  opelopabaf  4711