Theorem sbceq1a 3061

Description: Equality theorem for class substitution. Class version of sbequ12 1824. (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 1827	. 2 |- ( [ x / x ] ph <-> ph )
2		dfsbcq2 3054	. 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 1402   [wsb 1815   [.wsbc 3051

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-sbc 3052

This theorem is referenced by:  sbceq2a  3062  elrabsf  3090  cbvralcsf  3210  cbvrexcsf  3211  ifeqeqxdc  3684  rabsnifsb  3773  euotd  4390  omsinds  4764  elfvmptrab1  5794  ralrnmpt  5841  rexrnmpt  5842  riotass2  6057  riotass  6058  elovmporab  6279  elovmporab1w  6280  uchoice  6361  sbcopeq1a  6411  mpoxopoveq  6501  findcard2  7183  findcard2s  7184  ac6sfi  7192  opabfi  7237  dcfi  7305  indpi  7699  nn0ind-raph  9742  indstr  9972  fzrevral  10490  exfzdc  10637  zsupcllemstep  10640  infssuzex  10644  uzsinds  10859  wrdind  11472  wrd2ind  11473  prmind2  12876  gropd  16202  grstructd2dom  16203  bj-intabssel  16731  bj-bdfindes  16889  bj-findes  16921