Theorem sbceq1a 3008

Description: Equality theorem for class substitution. Class version of sbequ12 1794. (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 1797	. 2 |- ( [ x / x ] ph <-> ph )
2		dfsbcq2 3001	. 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 1373   [wsb 1785   [.wsbc 2998

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-sbc 2999

This theorem is referenced by:  sbceq2a  3009  elrabsf  3037  cbvralcsf  3156  cbvrexcsf  3157  euotd  4300  omsinds  4671  elfvmptrab1  5676  ralrnmpt  5724  rexrnmpt  5725  riotass2  5928  riotass  5929  elovmporab  6148  elovmporab1w  6149  uchoice  6225  sbcopeq1a  6275  mpoxopoveq  6328  findcard2  6988  findcard2s  6989  ac6sfi  6997  opabfi  7037  dcfi  7085  indpi  7457  nn0ind-raph  9492  indstr  9716  fzrevral  10229  exfzdc  10371  zsupcllemstep  10374  infssuzex  10378  uzsinds  10591  prmind2  12475  gropd  15677  grstructd2dom  15678  bj-intabssel  15762  bj-bdfindes  15922  bj-findes  15954