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Metamath Proof Explorer

Theorem eqnetrd

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

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

Proof

Step Hyp Ref Expression
1 eqnetrd.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 eqnetrd.2 ⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )
3 1 neeq1d ⊢ ( 𝜑 → ( 𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶 ) )
4 2 3 mpbird ⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )