Theorem pm13.181 3016

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 2745	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
2		pm13.18 3015	. 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)
3	1, 2	sylanb 584	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1542   ≠ wne 2934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-cleq 2730  df-ne 2935

This theorem is referenced by:  fzprval  13059  frgrwopreglem5a  28248  ax6e2ndeqALT  42089