Theorem sbceq1a 3055

Description: Equality theorem for class substitution. Class version of sbequ12 1820. (Contributed by NM, 26-Sep-2003.)

Assertion

Ref	Expression
sbceq1a	⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))

Proof of Theorem sbceq1a

Step	Hyp	Ref	Expression
1		sbid 1823	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
2		dfsbcq2 3048	. 2 ⊢ (𝑥 = 𝐴 → ([𝑥 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	1, 2	bitr3id 194	1 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  [wsb 1811  [wsbc 3045

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-sbc 3046

This theorem is referenced by:  sbceq2a  3056  elrabsf  3084  cbvralcsf  3204  cbvrexcsf  3205  ifeqeqxdc  3673  rabsnifsb  3762  euotd  4376  omsinds  4749  elfvmptrab1  5777  ralrnmpt  5824  rexrnmpt  5825  riotass2  6040  riotass  6041  elovmporab  6262  elovmporab1w  6263  uchoice  6344  sbcopeq1a  6394  mpoxopoveq  6484  findcard2  7159  findcard2s  7160  ac6sfi  7168  opabfi  7213  dcfi  7281  indpi  7673  nn0ind-raph  9713  indstr  9943  fzrevral  10461  exfzdc  10608  zsupcllemstep  10611  infssuzex  10615  uzsinds  10830  wrdind  11439  wrd2ind  11440  prmind2  12842  gropd  16168  grstructd2dom  16169  bj-intabssel  16687  bj-bdfindes  16845  bj-findes  16877