Theorem nonconne 2524

Description: Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)

Assertion

Ref	Expression
nonconne	⊢ ¬ (A = B ∧ A ≠ B)

Proof of Theorem nonconne

Step	Hyp	Ref	Expression
1		pm3.24 852	. 2 ⊢ ¬ (A = B ∧ ¬ A = B)
2		df-ne 2519	. . 3 ⊢ (A ≠ B ↔ ¬ A = B)
3	2	anbi2i 675	. 2 ⊢ ((A = B ∧ A ≠ B) ↔ (A = B ∧ ¬ A = B))
4	1, 3	mtbir 290	1 ⊢ ¬ (A = B ∧ A ≠ B)

Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ≠ wne 2517

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-an 360  df-ne 2519

This theorem is referenced by: (None)