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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