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Metamath Proof Explorer

Theorem neeqtrrd

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

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

Proof

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