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Metamath Proof Explorer

Theorem breqtrd

Description: Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999)

Ref Expression
Hypotheses breqtrd.1 ⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
breqtrd.2 ⊢ ( 𝜑 → 𝐵 = 𝐶 )
Assertion breqtrd ( 𝜑 → 𝐴 𝑅 𝐶 )

Proof

Step Hyp Ref Expression
1 breqtrd.1 ⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
2 breqtrd.2 ⊢ ( 𝜑 → 𝐵 = 𝐶 )
3 2 breq2d ⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ 𝐴 𝑅 𝐶 ) )
4 1 3 mpbid ⊢ ( 𝜑 → 𝐴 𝑅 𝐶 )