Theorem pm13.18 3014

Description: Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

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

Proof of Theorem pm13.18

Step	Hyp	Ref	Expression
1		eqeq1 2765	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
2	1	biimprd 238	. . 3 ⊢ (𝐴 = 𝐵 → (𝐵 = 𝐶 → 𝐴 = 𝐶))
3	2	necon3d 2954	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 → 𝐵 ≠ 𝐶))
4	3	imp 444	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ≠ wne 2933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-ext 2741

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-cleq 2754  df-ne 2934

This theorem is referenced by:  pm13.181  3015  frgrwopreglem5a  27487  4atexlemex4  35881  cncfiooicclem1  40628