Theorem sbceq1a 2995

Description: Equality theorem for class substitution. Class version of sbequ12 1782. (Contributed by NM, 26-Sep-2003.)

Assertion

Ref	Expression
sbceq1a	|- ( x = A -> ( ph <-> [. A / x ]. ph ) )

Proof of Theorem sbceq1a

Step	Hyp	Ref	Expression
1		sbid 1785	. 2 |- ( [ x / x ] ph <-> ph )
2		dfsbcq2 2988	. 2 |- ( x = A -> ( [ x / x ] ph <-> [. A / x ]. ph ) )
3	1, 2	bitr3id 194	1 |- ( x = A -> ( ph <-> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   [wsb 1773   [.wsbc 2985

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-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-sbc 2986

This theorem is referenced by:  sbceq2a  2996  elrabsf  3024  cbvralcsf  3143  cbvrexcsf  3144  euotd  4283  omsinds  4654  elfvmptrab1  5652  ralrnmpt  5700  rexrnmpt  5701  riotass2  5900  riotass  5901  elovmporab  6118  elovmporab1w  6119  uchoice  6190  sbcopeq1a  6240  mpoxopoveq  6293  findcard2  6945  findcard2s  6946  ac6sfi  6954  opabfi  6992  dcfi  7040  indpi  7402  nn0ind-raph  9434  indstr  9658  fzrevral  10171  exfzdc  10307  uzsinds  10515  zsupcllemstep  12082  infssuzex  12086  prmind2  12258  bj-intabssel  15281  bj-bdfindes  15441  bj-findes  15473