Theorem sbc2ie 3114

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 1619	. . 3 |- F/xps
4		nfv 1619	. . 3 |- F/yps
5	2	nfth 1553	. . 3 |- F/x B e. _V
6		sbc2ie.3	. . 3 |- ((x = A /\ y = B) -> (ph <-> ps))
7	3, 4, 5, 6	sbc2iegf 3113	. 2 |- ((A e. _V /\ B e. _V) -> ( [.A / x ]. [.B / y ].ph <-> ps))
8	1, 2, 7	mp2an 653	1 |- ( [.A / x ]. [.B / y ].ph <-> ps)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  _Vcvv 2860   [.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:  sbc3ie  3116