Theorem exmidne 2522

Description: Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.)

Assertion

Ref	Expression
exmidne	⊢ (A = B ∨ A ≠ B)

Proof of Theorem exmidne

Step	Hyp	Ref	Expression
1		exmid 404	. 2 ⊢ (A = B ∨ ¬ A = B)
2		df-ne 2518	. . 3 ⊢ (A ≠ B ↔ ¬ A = B)
3	2	orbi2i 505	. 2 ⊢ ((A = B ∨ A ≠ B) ↔ (A = B ∨ ¬ A = B))
4	1, 3	mpbir 200	1 ⊢ (A = B ∨ A ≠ B)

Syntax hints:  ¬ wn 3   ∨ wo 357   = wceq 1642   ≠ wne 2516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-or 359  df-ne 2518

This theorem is referenced by: (None)