Theorem nonconne 2379

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		pm3.24 694	. 2 ⊢ ¬ (𝐴 = 𝐵 ∧ ¬ 𝐴 = 𝐵)
2		df-ne 2368	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	2	anbi2i 457	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵) ↔ (𝐴 = 𝐵 ∧ ¬ 𝐴 = 𝐵))
4	1, 3	mtbir 672	1 ⊢ ¬ (𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1364   ≠ wne 2367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117  df-ne 2368

This theorem is referenced by: (None)