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Metamath Proof Explorer

Theorem eqnbrtrd

Description: Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021)

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

Proof

Step Hyp Ref Expression
1 eqnbrtrd.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 eqnbrtrd.2 ⊢ ( 𝜑 → ¬ 𝐵 𝑅 𝐶 )
3 1 breq1d ⊢ ( 𝜑 → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐶 ) )
4 2 3 mtbird ⊢ ( 𝜑 → ¬ 𝐴 𝑅 𝐶 )