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Metamath Proof Explorer

Theorem breqtrrid

Description: A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005)

Ref Expression
Hypotheses breqtrrid.1 ⊢ A R B
breqtrrid.2 ⊢ φ → C = B
Assertion breqtrrid ⊢ φ → A R C

Proof

Step Hyp Ref Expression
1 breqtrrid.1 ⊢ A R B
2 breqtrrid.2 ⊢ φ → C = B
3 2 eqcomd ⊢ φ → B = C
4 1 3 breqtrid ⊢ φ → A R C