Theorem sbcie2g 2984

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2985 avoids a disjointness condition on x and 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 2953	. 2 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
2		sbcie2g.2	. 2 |- ( y = A -> ( ps <-> ch ) )
3		sbsbc 2955	. . 3 |- ( [ y / x ] ph <-> [. y / x ]. ph )
4		nfv 1516	. . . 4 |- F/ x ps
5		sbcie2g.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
6	4, 5	sbie 1779	. . 3 |- ( [ y / x ] ph <-> ps )
7	3, 6	bitr3i 185	. 2 |- ( [. y / x ]. ph <-> ps )
8	1, 2, 7	vtoclbg 2787	1 |- ( A e. V -> ( [. A / x ]. ph <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   [wsb 1750   e. wcel 2136   [.wsbc 2951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952

This theorem is referenced by:  sbcel2gv  3014  csbie2g  3095  brab1  4029