Theorem releqi 4441

Description: Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)

Hypothesis

Ref	Expression
releqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
releqi	⊢ (Rel 𝐴 ↔ Rel 𝐵)

Proof of Theorem releqi

Step	Hyp	Ref	Expression
1		releqi.1	. 2 ⊢ 𝐴 = 𝐵
2		releq 4440	. 2 ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))
3	1, 2	ax-mp 7	1 ⊢ (Rel 𝐴 ↔ Rel 𝐵)

Syntax hints:   ↔ wb 103   = wceq 1284  Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986  df-rel 4370

This theorem is referenced by:  reliun  4476  reluni  4478  relint  4479  reldmmpt2  5632  tfrlem6  5955