Theorem pm13.181OLD 3026

Description: Obsolete version of pm13.181 3025 as of 30-Oct-2024. (Contributed by Andrew Salmon, 3-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem pm13.181OLD

Step	Hyp	Ref	Expression
1		eqcom 2745	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
2		pm13.18 3024	. 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)
3	1, 2	sylanb 580	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ≠ wne 2942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-cleq 2730  df-ne 2943

This theorem is referenced by: (None)