Metamath Proof Explorer

Theorem eqeq1

Description: Equality implies equivalence of equalities. (Contributed by NM, 26-May-1993) (Proof shortened by Wolf Lammen, 19-Nov-2019)

Ref Expression
Assertion eqeq1 ( 𝐴 = 𝐵 → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐶 ) )

Proof

Step Hyp Ref Expression
1 id ⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 )
2 1 eqeq1d ⊢ ( 𝐴 = 𝐵 → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐶 ) )