Theorem nonconne 2352

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		pm3.24 688	. 2 ⊢ ¬ (𝐴 = 𝐵 ∧ ¬ 𝐴 = 𝐵)
2		df-ne 2341	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	2	anbi2i 454	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) ↔ (𝐴 = 𝐵 ∧ ¬ 𝐴 = 𝐵))
4	1, 3	mtbir 666	1 ⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 103   = wceq 1348   ≠ wne 2340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610

This theorem depends on definitions:  df-bi 116  df-ne 2341

This theorem is referenced by: (None)