Theorem sbceq1a 2918

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 2912	. 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 2909

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 2121

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-sbc 2910

This theorem is referenced by:  sbceq2a  2919  elrabsf  2947  cbvralcsf  3062  cbvrexcsf  3063  euotd  4176  omsinds  4535  elfvmptrab1  5515  ralrnmpt  5562  rexrnmpt  5563  riotass2  5756  riotass  5757  sbcopeq1a  6085  mpoxopoveq  6137  findcard2  6783  findcard2s  6784  ac6sfi  6792  indpi  7150  nn0ind-raph  9168  indstr  9388  fzrevral  9885  exfzdc  10017  uzsinds  10215  zsupcllemstep  11638  infssuzex  11642  prmind2  11801  bj-intabssel  12996  bj-bdfindes  13147  bj-findes  13179