Theorem pm13.181

Description: Theorem *13.181 in WhiteheadRussell p. 178. (Contributed by Andrew Salmon, 3-Jun-2011)

	Ref	Expression
Assertion	pm13.181	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶 ) → 𝐴 ≠ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eqcom	⊢ ( 𝐴 = 𝐵 ↔ 𝐵 = 𝐴 )
2		pm13.18	⊢ ( ( 𝐵 = 𝐴 ∧ 𝐵 ≠ 𝐶 ) → 𝐴 ≠ 𝐶 )
3	1 2	sylanb	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶 ) → 𝐴 ≠ 𝐶 )