Theorem sbceq1a 2999

Description: Equality theorem for class substitution. Class version of sbequ12 1785. (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 1788	. 2 |- ( [ x / x ] ph <-> ph )
2		dfsbcq2 2992	. 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 1364   [wsb 1776   [.wsbc 2989

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-sbc 2990

This theorem is referenced by:  sbceq2a  3000  elrabsf  3028  cbvralcsf  3147  cbvrexcsf  3148  euotd  4287  omsinds  4658  elfvmptrab1  5656  ralrnmpt  5704  rexrnmpt  5705  riotass2  5904  riotass  5905  elovmporab  6123  elovmporab1w  6124  uchoice  6195  sbcopeq1a  6245  mpoxopoveq  6298  findcard2  6950  findcard2s  6951  ac6sfi  6959  opabfi  6999  dcfi  7047  indpi  7409  nn0ind-raph  9443  indstr  9667  fzrevral  10180  exfzdc  10316  zsupcllemstep  10319  infssuzex  10323  uzsinds  10536  prmind2  12288  bj-intabssel  15435  bj-bdfindes  15595  bj-findes  15627