Theorem sbceq1d 2960

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

Hypothesis

Ref	Expression
sbceq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
sbceq1d	|- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ps ) )

Proof of Theorem sbceq1d

Step	Hyp	Ref	Expression
1		sbceq1d.1	. 2 |- ( ph -> A = B )
2		dfsbcq 2957	. 2 |- ( A = B -> ( [. A / x ]. ps <-> [. B / x ]. ps ) )
3	1, 2	syl 14	1 |- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ps ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   [.wsbc 2955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166  df-sbc 2956

This theorem is referenced by:  sbceq1dd  2961  rexrnmpt  5637  findcard2  6864  findcard2s  6865  ac6sfi  6873  nn1suc  8886  uzind4s  9538  uzind4s2  9539  fzrevral  10050  fzshftral  10053  cjth  10799  prmind2  12063  bj-bdfindes  13946  bj-findes  13978