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Metamath Proof Explorer

Theorem eqnetrrid

Description: A chained equality inference for inequality. (Contributed by NM, 6-Jun-2012) (Proof shortened by Wolf Lammen, 19-Nov-2019)

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

Proof

Step Hyp Ref Expression
1 eqnetrrid.1 ⊢ 𝐵 = 𝐴
2 eqnetrrid.2 ⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )
3 1 a1i ⊢ ( 𝜑 → 𝐵 = 𝐴 )
4 3 2 eqnetrrd ⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )