Theorem sbcie2g 3080

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3081 avoids a disjointness condition on x, A by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
sbcie2g.1	|- (x = y -> (ph <-> ps))
sbcie2g.2	|- (y = A -> (ps <-> ch))

Assertion

Ref	Expression
sbcie2g	|- (A e. V -> ( [.A / x ].ph <-> ch))

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

Allowed substitution hints:   ph(x)   ps(y)   ch(x)   A(x)   V(x,y)

Proof of Theorem sbcie2g

Step	Hyp	Ref	Expression
1		dfsbcq 3049	. 2 |- (y = A -> ( [.y / x ].ph <-> [.A / x ].ph))
2		sbcie2g.2	. 2 |- (y = A -> (ps <-> ch))
3		sbsbc 3051	. . 3 |- ([y / x]ph <-> [.y / x ].ph)
4		nfv 1619	. . . 4 |- F/xps
5		sbcie2g.1	. . . 4 |- (x = y -> (ph <-> ps))
6	4, 5	sbie 2038	. . 3 |- ([y / x]ph <-> ps)
7	3, 6	bitr3i 242	. 2 |- ( [.y / x ].ph <-> ps)
8	1, 2, 7	vtoclbg 2916	1 |- (A e. V -> ( [.A / x ].ph <-> ch))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642  [wsb 1648   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-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:  csbie2g  3183  brab1  4685