Theorem pm13.18

Description: Theorem *13.18 in WhiteheadRussell p. 178. (Contributed by Andrew Salmon, 3-Jun-2011) (Proof shortened by Wolf Lammen, 14-May-2023)

	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	biimpd	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ≠ 𝐶 → 𝐵 ≠ 𝐶 ) )
3	2	imp	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶 ) → 𝐵 ≠ 𝐶 )