Theorem sbceq1a 3038

Description: Equality theorem for class substitution. Class version of sbequ12 1817. (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 1820	. 2 |- ( [ x / x ] ph <-> ph )
2		dfsbcq2 3031	. 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 1395   [wsb 1808   [.wsbc 3028

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-sbc 3029

This theorem is referenced by:  sbceq2a  3039  elrabsf  3067  cbvralcsf  3187  cbvrexcsf  3188  euotd  4341  omsinds  4714  elfvmptrab1  5729  ralrnmpt  5777  rexrnmpt  5778  riotass2  5983  riotass  5984  elovmporab  6205  elovmporab1w  6206  uchoice  6283  sbcopeq1a  6333  mpoxopoveq  6386  findcard2  7051  findcard2s  7052  ac6sfi  7060  opabfi  7100  dcfi  7148  indpi  7529  nn0ind-raph  9564  indstr  9788  fzrevral  10301  exfzdc  10446  zsupcllemstep  10449  infssuzex  10453  uzsinds  10666  wrdind  11254  wrd2ind  11255  prmind2  12642  gropd  15848  grstructd2dom  15849  bj-intabssel  16153  bj-bdfindes  16312  bj-findes  16344