Theorem sbcie2g 3032

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3033 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 3000	. 2 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
2		sbcie2g.2	. 2 |- ( y = A -> ( ps <-> ch ) )
3		sbsbc 3002	. . 3 |- ( [ y / x ] ph <-> [. y / x ]. ph )
4		nfv 1551	. . . 4 |- F/ x ps
5		sbcie2g.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
6	4, 5	sbie 1814	. . 3 |- ( [ y / x ] ph <-> ps )
7	3, 6	bitr3i 186	. 2 |- ( [. y / x ]. ph <-> ps )
8	1, 2, 7	vtoclbg 2834	1 |- ( A e. V -> ( [. A / x ]. ph <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   [wsb 1785   e. wcel 2176   [.wsbc 2998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sbc 2999

This theorem is referenced by:  sbcel2gv  3062  csbie2g  3144  brab1  4091