Theorem sbceq1a 2922

Description: Equality theorem for class substitution. Class version of sbequ12 1745. (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 1748	. 2 |- ( [ x / x ] ph <-> ph )
2		dfsbcq2 2916	. 2 |- ( x = A -> ( [ x / x ] ph <-> [. A / x ]. ph ) )
3	1, 2	bitr3id 193	1 |- ( x = A -> ( ph <-> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   [wsb 1736   [.wsbc 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-sbc 2914

This theorem is referenced by:  sbceq2a  2923  elrabsf  2951  cbvralcsf  3067  cbvrexcsf  3068  euotd  4184  omsinds  4543  elfvmptrab1  5523  ralrnmpt  5570  rexrnmpt  5571  riotass2  5764  riotass  5765  sbcopeq1a  6093  mpoxopoveq  6145  findcard2  6791  findcard2s  6792  ac6sfi  6800  indpi  7174  nn0ind-raph  9192  indstr  9415  fzrevral  9916  exfzdc  10048  uzsinds  10246  zsupcllemstep  11674  infssuzex  11678  prmind2  11837  bj-intabssel  13167  bj-bdfindes  13318  bj-findes  13350