This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem breqdi

Description: Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020)

Ref Expression
Hypotheses breq1d.1 ⊢ φ → A = B
breqdi.1 ⊢ φ → C A D
Assertion breqdi ⊢ φ → C B D

Proof

Step Hyp Ref Expression
1 breq1d.1 ⊢ φ → A = B
2 breqdi.1 ⊢ φ → C A D
3 1 breqd ⊢ φ → C A D ↔ C B D
4 2 3 mpbid ⊢ φ → C B D