Theorem sbceq1d 2820

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 2817	. 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 103   = wceq 1284   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077  df-sbc 2816

This theorem is referenced by:  sbceq1dd  2821  rexrnmpt  5331  findcard2  6373  findcard2s  6374  ac6sfi  6379  nn1suc  8058  uzind4s  8678  uzind4s2  8679  fzrevral  9122  fzshftral  9125  cjth  9733  prmind2  10502  bj-bdfindes  10744  bj-findes  10776