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Metamath Proof Explorer

Theorem pm13.18

Description: Theorem *13.18 in WhiteheadRussell p. 178. (Contributed by Andrew Salmon, 3-Jun-2011) (Proof shortened by Wolf Lammen, 29-Oct-2024)

Ref Expression
Assertion pm13.18 ( ( 𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶 ) → 𝐵 ≠ 𝐶 )

Proof

Step Hyp Ref Expression
1 neeq1 ⊢ ( 𝐴 = 𝐵 → ( 𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶 ) )
2 1 biimpa ⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶 ) → 𝐵 ≠ 𝐶 )