Theorem sbcied2 3012

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 2242	. . 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 3011	1 |- ( ph -> ( [. A / x ]. ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1363   e. wcel 2158   [.wsbc 2974

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-sbc 2975

This theorem is referenced by:  ismgm  12795  issgrp  12828  isnsg  13094  isrng  13186  isring  13252