Theorem sbceq1d 2982

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 2979	. 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 105   = wceq 1364   [.wsbc 2977

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185  df-sbc 2978

This theorem is referenced by:  sbceq1dd  2983  rexrnmpt  5680  findcard2  6917  findcard2s  6918  ac6sfi  6926  nn1suc  8968  uzind4s  9620  uzind4s2  9621  fzrevral  10135  fzshftral  10138  cjth  10887  prmind2  12152  issrg  13319  islmod  13607  bj-bdfindes  15162  bj-findes  15194