Theorem nonconne 2267

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		pm3.24 662	. 2 ⊢ ¬ (𝐴 = 𝐵 ∧ ¬ 𝐴 = 𝐵)
2		df-ne 2256	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	2	anbi2i 445	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) ↔ (𝐴 = 𝐵 ∧ ¬ 𝐴 = 𝐵))
4	1, 3	mtbir 631	1 ⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 102   = wceq 1289   ≠ wne 2255

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580

This theorem depends on definitions:  df-bi 115  df-ne 2256

This theorem is referenced by: (None)