Theorem sbcied2 3066

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)

Hypotheses

Ref	Expression
sbcied2.1	|- ( ph -> A e. V )
sbcied2.2	|- ( ph -> A = B )
sbcied2.3	|- ( ( ph /\ x = B ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
sbcied2	|- ( ph -> ( [. A / x ]. ps <-> ch ) )

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

Allowed substitution hints:   ps( x)   B( x)   V( x)

Proof of Theorem sbcied2

Step	Hyp	Ref	Expression
1		sbcied2.1	. 2 |- ( ph -> A e. V )
2		id 19	. . . 4 |- ( x = A -> x = A )
3		sbcied2.2	. . . 4 |- ( ph -> A = B )
4	2, 3	sylan9eqr 2284	. . 3 |- ( ( ph /\ x = A ) -> x = B )
5		sbcied2.3	. . 3 |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )
6	4, 5	syldan 282	. 2 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
7	1, 6	sbcied 3065	1 |- ( ph -> ( [. A / x ]. ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   [.wsbc 3028

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029

This theorem is referenced by:  ismgm  13390  issgrp  13436  isnsg  13739  isrng  13897  isring  13963  isdomn  14233  isuhgrm  15871  isushgrm  15872  isupgren  15895  isumgren  15905  isuspgren  15955  isusgren  15956