Theorem sbceq1a 2972

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 2965	. 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 2962

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 2963

This theorem is referenced by:  sbceq2a  2973  elrabsf  3001  cbvralcsf  3119  cbvrexcsf  3120  euotd  4254  omsinds  4621  elfvmptrab1  5610  ralrnmpt  5658  rexrnmpt  5659  riotass2  5856  riotass  5857  sbcopeq1a  6187  mpoxopoveq  6240  findcard2  6888  findcard2s  6889  ac6sfi  6897  dcfi  6979  indpi  7340  nn0ind-raph  9369  indstr  9592  fzrevral  10104  exfzdc  10239  uzsinds  10441  zsupcllemstep  11945  infssuzex  11949  prmind2  12119  bj-intabssel  14511  bj-bdfindes  14671  bj-findes  14703