Theorem sbceq1a 2973

Description: Equality theorem for class substitution. Class version of sbequ12 1771. (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 1774	. 2 |- ( [ x / x ] ph <-> ph )
2		dfsbcq2 2966	. 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 1353   [wsb 1762   [.wsbc 2963

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-sbc 2964

This theorem is referenced by:  sbceq2a  2974  elrabsf  3002  cbvralcsf  3120  cbvrexcsf  3121  euotd  4255  omsinds  4622  elfvmptrab1  5611  ralrnmpt  5659  rexrnmpt  5660  riotass2  5857  riotass  5858  sbcopeq1a  6188  mpoxopoveq  6241  findcard2  6889  findcard2s  6890  ac6sfi  6898  dcfi  6980  indpi  7341  nn0ind-raph  9370  indstr  9593  fzrevral  10105  exfzdc  10240  uzsinds  10442  zsupcllemstep  11946  infssuzex  11950  prmind2  12120  bj-intabssel  14544  bj-bdfindes  14704  bj-findes  14736