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 ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   [wsb 1815   [.wsbc 3051

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

This theorem is used by:  sbceq2a  3062  elrabsf  3090  cbvralcsf  3210  cbvrexcsf  3211  ifeqeqxdc  3687  rabsnifsb  3777  euotd  4395  omsinds  4769  elfvmptrab1  5801  ralrnmpt  5850  rexrnmpt  5851  riotass2  6067  riotass  6068  elovmporab  6289  elovmporab1w  6290  uchoice  6371  sbcopeq1a  6421  mpoxopoveq  6511  findcard2  7193  findcard2s  7194  ac6sfi  7202  opabfi  7247  dcfi  7315  indpi  7709  nn0ind-raph  9767  indstr  10002  fzrevral  10522  exfzdc  10669  zsupcllemstep  10672  infssuzex  10676  uzsinds  10894  wrdind  11508  wrd2ind  11509  prmind2  12914  gropd  16386  grstructd2dom  16387  bj-intabssel  16915  bj-bdfindes  17073  bj-findes  17105