Theorem sbcied2 3027

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 2251	. . 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 3026	1 |- ( ph -> ( [. A / x ]. ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   [.wsbc 2989

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990

This theorem is referenced by:  ismgm  13000  issgrp  13046  isnsg  13332  isrng  13490  isring  13556  isdomn  13825