Theorem sbceq1a 2960

Description: Equality theorem for class substitution. Class version of sbequ12 1759. (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 1762	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
2		dfsbcq2 2954	. 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 1343  [wsb 1750  [wsbc 2951

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-sbc 2952

This theorem is referenced by:  sbceq2a  2961  elrabsf  2989  cbvralcsf  3107  cbvrexcsf  3108  euotd  4232  omsinds  4599  elfvmptrab1  5580  ralrnmpt  5627  rexrnmpt  5628  riotass2  5824  riotass  5825  sbcopeq1a  6155  mpoxopoveq  6208  findcard2  6855  findcard2s  6856  ac6sfi  6864  dcfi  6946  indpi  7283  nn0ind-raph  9308  indstr  9531  fzrevral  10040  exfzdc  10175  uzsinds  10377  zsupcllemstep  11878  infssuzex  11882  prmind2  12052  bj-intabssel  13670  bj-bdfindes  13831  bj-findes  13863