Theorem pm13.181 2446

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 2195	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
2		pm13.18 2445	. 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)
3	1, 2	sylanb 284	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ≠ wne 2364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-ne 2365

This theorem is referenced by:  nninfisollemne  7190  fzprval  10148  mod2eq1n2dvds  12020