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Metamath Proof Explorer

Theorem eqbrtrdi

Description: A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999)

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

Proof

Step Hyp Ref Expression
1 eqbrtrdi.1 ⊢ φ → A = B
2 eqbrtrdi.2 ⊢ B R C
3 1 breq1d ⊢ φ → A R C ↔ B R C
4 2 3 mpbiri ⊢ φ → A R C