Theorem pm13.181 2331

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

Assertion

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

Proof of Theorem pm13.181

Step	Hyp	Ref	Expression
1		eqcom 2085	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
2		pm13.18 2330	. 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)
3	1, 2	sylanb 278	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1285   ≠ wne 2249

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-cleq 2076  df-ne 2250

This theorem is referenced by:  fzprval  9211  mod2eq1n2dvds  10470