Theorem sbc2iedv 3115

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Hypotheses

Ref	Expression
sbc2iedv.1	|- A e. _V
sbc2iedv.2	|- B e. _V
sbc2iedv.3	|- (ph -> ((x = A /\ y = B) -> (ps <-> ch)))

Assertion

Ref	Expression
sbc2iedv	|- (ph -> ( [.A / x ]. [.B / y ].ps <-> ch))

Distinct variable groups:   x,y,A   y,B   ph,x,y   ch,x,y

Allowed substitution hints:   ps(x,y)   B(x)

Proof of Theorem sbc2iedv

Step	Hyp	Ref	Expression
1		sbc2iedv.1	. . 3 |- A e. _V
2	1	a1i 10	. 2 |- (ph -> A e. _V)
3		sbc2iedv.2	. . . 4 |- B e. _V
4	3	a1i 10	. . 3 |- ((ph /\ x = A) -> B e. _V)
5		sbc2iedv.3	. . . 4 |- (ph -> ((x = A /\ y = B) -> (ps <-> ch)))
6	5	impl 603	. . 3 |- (((ph /\ x = A) /\ y = B) -> (ps <-> ch))
7	4, 6	sbcied 3083	. 2 |- ((ph /\ x = A) -> ( [.B / y ].ps <-> ch))
8	2, 7	sbcied 3083	1 |- (ph -> ( [.A / x ]. [.B / y ].ps <-> ch))

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: (None)