Theorem sbceq1a 3008

Description: Equality theorem for class substitution. Class version of sbequ12 1794. (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 1797	. 2 |- ( [ x / x ] ph <-> ph )
2		dfsbcq2 3001	. 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 1785   [.wsbc 2998

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-sbc 2999

This theorem is referenced by:  sbceq2a  3009  elrabsf  3037  cbvralcsf  3156  cbvrexcsf  3157  euotd  4299  omsinds  4670  elfvmptrab1  5674  ralrnmpt  5722  rexrnmpt  5723  riotass2  5926  riotass  5927  elovmporab  6146  elovmporab1w  6147  uchoice  6223  sbcopeq1a  6273  mpoxopoveq  6326  findcard2  6986  findcard2s  6987  ac6sfi  6995  opabfi  7035  dcfi  7083  indpi  7455  nn0ind-raph  9490  indstr  9714  fzrevral  10227  exfzdc  10369  zsupcllemstep  10372  infssuzex  10376  uzsinds  10589  prmind2  12442  gropd  15644  grstructd2dom  15645  bj-intabssel  15725  bj-bdfindes  15885  bj-findes  15917