Theorem sbceq1a 3015

Description: Equality theorem for class substitution. Class version of sbequ12 1795. (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 1798	. 2 |- ( [ x / x ] ph <-> ph )
2		dfsbcq2 3008	. 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 1373   [wsb 1786   [.wsbc 3005

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-sbc 3006

This theorem is referenced by:  sbceq2a  3016  elrabsf  3044  cbvralcsf  3164  cbvrexcsf  3165  euotd  4317  omsinds  4688  elfvmptrab1  5697  ralrnmpt  5745  rexrnmpt  5746  riotass2  5949  riotass  5950  elovmporab  6169  elovmporab1w  6170  uchoice  6246  sbcopeq1a  6296  mpoxopoveq  6349  findcard2  7012  findcard2s  7013  ac6sfi  7021  opabfi  7061  dcfi  7109  indpi  7490  nn0ind-raph  9525  indstr  9749  fzrevral  10262  exfzdc  10406  zsupcllemstep  10409  infssuzex  10413  uzsinds  10626  wrdind  11213  wrd2ind  11214  prmind2  12557  gropd  15761  grstructd2dom  15762  bj-intabssel  15925  bj-bdfindes  16084  bj-findes  16116