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Metamath Proof Explorer

Theorem breq12i

Description: Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996) (Proof shortened by Eric Schmidt, 4-Apr-2007)

Ref Expression
Hypotheses breq1i.1 ⊢ 𝐴 = 𝐵
breq12i.2 ⊢ 𝐶 = 𝐷
Assertion breq12i ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 )

Proof

Step Hyp Ref Expression
1 breq1i.1 ⊢ 𝐴 = 𝐵
2 breq12i.2 ⊢ 𝐶 = 𝐷
3 breq12 ⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 ) )
4 1 2 3 mp2an ⊢ ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 )