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

Metamath Proof Explorer

Theorem breq2dd

Description: Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 10-Jan-2026)

Ref Expression
Hypotheses breq2dd.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
breq2dd.2 ⊢ ( 𝜑 → 𝐶 𝑅 𝐴 )
Assertion breq2dd ( 𝜑 → 𝐶 𝑅 𝐵 )

Proof

Step Hyp Ref Expression
1 breq2dd.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 breq2dd.2 ⊢ ( 𝜑 → 𝐶 𝑅 𝐴 )
3 1 breq2d ⊢ ( 𝜑 → ( 𝐶 𝑅 𝐴 ↔ 𝐶 𝑅 𝐵 ) )
4 2 3 mpbid ⊢ ( 𝜑 → 𝐶 𝑅 𝐵 )