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Metamath Proof Explorer

Theorem 3netr4g

Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012)

Ref Expression
Hypotheses 3netr4g.1 ⊢ φ → A ≠ B
3netr4g.2 ⊢ C = A
3netr4g.3 ⊢ D = B
Assertion 3netr4g ⊢ φ → C ≠ D

Proof

Step Hyp Ref Expression
1 3netr4g.1 ⊢ φ → A ≠ B
2 3netr4g.2 ⊢ C = A
3 3netr4g.3 ⊢ D = B
4 2 3 neeq12i ⊢ C ≠ D ↔ A ≠ B
5 1 4 sylibr ⊢ φ → C ≠ D