Theorem sbcie2g 3469

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3470 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 3437	. 2 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
2		sbcie2g.2	. 2 |- ( y = A -> ( ps <-> ch ) )
3		sbsbc 3439	. . 3 |- ( [ y / x ] ph <-> [. y / x ]. ph )
4		nfv 1843	. . . 4 |- F/ x ps
5		sbcie2g.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
6	4, 5	sbie 2408	. . 3 |- ( [ y / x ] ph <-> ps )
7	3, 6	bitr3i 266	. 2 |- ( [. y / x ]. ph <-> ps )
8	1, 2, 7	vtoclbg 3267	1 |- ( A e. V -> ( [. A / x ]. ph <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   [wsb 1880   e. wcel 1990   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  sbcel2gv  3496  csbie2g  3564  brab1  4700  bnj90  30788  bnj124  30941  riotasvd  34242