Theorem sbcie2g 2946

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2947 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 2915	. 2 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
2		sbcie2g.2	. 2 |- ( y = A -> ( ps <-> ch ) )
3		sbsbc 2917	. . 3 |- ( [ y / x ] ph <-> [. y / x ]. ph )
4		nfv 1509	. . . 4 |- F/ x ps
5		sbcie2g.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
6	4, 5	sbie 1765	. . 3 |- ( [ y / x ] ph <-> ps )
7	3, 6	bitr3i 185	. 2 |- ( [. y / x ]. ph <-> ps )
8	1, 2, 7	vtoclbg 2750	1 |- ( A e. V -> ( [. A / x ]. ph <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   e. wcel 1481   [wsb 1736   [.wsbc 2913

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2914

This theorem is referenced by:  sbcel2gv  2976  csbie2g  3055  brab1  3983