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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