Theorem sbceq1a 2946

Description: Equality theorem for class substitution. Class version of sbequ12 1751. (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 1754	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
2		dfsbcq2 2940	. 2 ⊢ (𝑥 = 𝐴 → ([𝑥 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	1, 2	bitr3id 193	1 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1335  [wsb 1742  [wsbc 2937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-sbc 2938

This theorem is referenced by:  sbceq2a  2947  elrabsf  2975  cbvralcsf  3093  cbvrexcsf  3094  euotd  4214  omsinds  4580  elfvmptrab1  5561  ralrnmpt  5608  rexrnmpt  5609  riotass2  5803  riotass  5804  sbcopeq1a  6132  mpoxopoveq  6184  findcard2  6831  findcard2s  6832  ac6sfi  6840  indpi  7256  nn0ind-raph  9275  indstr  9498  fzrevral  10000  exfzdc  10132  uzsinds  10334  zsupcllemstep  11824  infssuzex  11828  prmind2  11988  bj-intabssel  13334  bj-bdfindes  13495  bj-findes  13527