Theorem pm13.181 2391

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 2142	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
2		pm13.18 2390	. 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)
3	1, 2	sylanb 282	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ≠ wne 2309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-4 1488  ax-17 1507  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-ne 2310

This theorem is referenced by:  fzprval  9893  mod2eq1n2dvds  11612