Theorem sbceq1dd 2970

Description: Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)

Hypotheses

Ref	Expression
sbceq1d.1	|- ( ph -> A = B )
sbceq1dd.2	|- ( ph -> [. A / x ]. ps )

Assertion

Ref	Expression
sbceq1dd	|- ( ph -> [. B / x ]. ps )

Proof of Theorem sbceq1dd

Step	Hyp	Ref	Expression
1		sbceq1dd.2	. 2 |- ( ph -> [. A / x ]. ps )
2		sbceq1d.1	. . 3 |- ( ph -> A = B )
3	2	sbceq1d 2969	. 2 |- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ps ) )
4	1, 3	mpbid 147	1 |- ( ph -> [. B / x ]. ps )

Syntax hints:   -> wi 4   = wceq 1353   [.wsbc 2964

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-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-sbc 2965

This theorem is referenced by:  prmind2  12122