Theorem nancom 1491

Description: Alternative denial is commutative. Remark: alternative denial is not associative, see nanass 1505. (Contributed by Mario Carneiro, 9-May-2015.) (Proof shortened by Wolf Lammen, 26-Jun-2020.)

Assertion

Ref	Expression
nancom	⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜑))

Proof of Theorem nancom

Step	Hyp	Ref	Expression
1		con2b 360	. 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))
2		dfnan2 1489	. 2 ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓))
3		dfnan2 1489	. 2 ⊢ ((𝜓 ⊼ 𝜑) ↔ (𝜓 → ¬ 𝜑))
4	1, 2, 3	3bitr4i 303	1 ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ⊼ wnan 1486

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 206  df-an 397  df-nan 1487

This theorem is referenced by:  nanbi2  1497  nanass  1505  falnantru  1582