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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 )