Theorem sbceq1a 2942

Description: Equality theorem for class substitution. Class version of sbequ12 1748. (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 1751	. 2 |- ( [ x / x ] ph <-> ph )
2		dfsbcq2 2936	. 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 1739   [.wsbc 2933

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 1481  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-sbc 2934

This theorem is referenced by:  sbceq2a  2943  elrabsf  2971  cbvralcsf  3089  cbvrexcsf  3090  euotd  4209  omsinds  4575  elfvmptrab1  5555  ralrnmpt  5602  rexrnmpt  5603  riotass2  5796  riotass  5797  sbcopeq1a  6125  mpoxopoveq  6177  findcard2  6823  findcard2s  6824  ac6sfi  6832  indpi  7241  nn0ind-raph  9260  indstr  9483  fzrevral  9985  exfzdc  10117  uzsinds  10319  zsupcllemstep  11805  infssuzex  11809  prmind2  11968  bj-intabssel  13301  bj-bdfindes  13462  bj-findes  13494