Theorem sbceq1d 2914

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 2911	. 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 1331   [.wsbc 2909

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135  df-sbc 2910

This theorem is referenced by:  sbceq1dd  2915  rexrnmpt  5563  findcard2  6783  findcard2s  6784  ac6sfi  6792  nn1suc  8739  uzind4s  9385  uzind4s2  9386  fzrevral  9885  fzshftral  9888  cjth  10618  prmind2  11801  bj-bdfindes  13147  bj-findes  13179