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Metamath Proof Explorer

Theorem eqnetrrd

Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012)

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

Proof

Step Hyp Ref Expression
1 eqnetrrd.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 eqnetrrd.2 ⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )
3 1 eqcomd ⊢ ( 𝜑 → 𝐵 = 𝐴 )
4 3 2 eqnetrd ⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )