Theorem pm13.18 3025

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

Assertion

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

Proof of Theorem pm13.18

Step	Hyp	Ref	Expression
1		neeq1 3006	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
2	1	biimpa 477	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ≠ wne 2943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2730  df-ne 2944

This theorem is referenced by:  pm13.181OLD  3027  iotan0  6423  frgrwopreglem5a  28675  4atexlemex4  38087  cncfiooicclem1  43434