Theorem sbceq1a 2913

Description: Equality theorem for class substitution. Class version of sbequ12 1744. (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 1747	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
2		dfsbcq2 2907	. 2 ⊢ (𝑥 = 𝐴 → ([𝑥 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	1, 2	syl5bbr 193	1 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331  [wsb 1735  [wsbc 2904

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-sbc 2905

This theorem is referenced by:  sbceq2a  2914  elrabsf  2942  cbvralcsf  3057  cbvrexcsf  3058  euotd  4171  omsinds  4530  elfvmptrab1  5508  ralrnmpt  5555  rexrnmpt  5556  riotass2  5749  riotass  5750  sbcopeq1a  6078  mpoxopoveq  6130  findcard2  6776  findcard2s  6777  ac6sfi  6785  indpi  7143  nn0ind-raph  9161  indstr  9381  fzrevral  9878  exfzdc  10010  uzsinds  10208  zsupcllemstep  11627  infssuzex  11631  prmind2  11790  bj-intabssel  12985  bj-bdfindes  13136  bj-findes  13168