Theorem wl-nancom 32476

Description: The 'nand' operator commutes. (Contributed by Mario Carneiro, 9-May-2015.) (Revised by Wolf Lammen, 26-Jun-2020.)

Assertion

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

Proof of Theorem wl-nancom

Step	Hyp	Ref	Expression
1		con2b 348	. 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))
2		wl-dfnan2 32475	. 2 ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓))
3		wl-dfnan2 32475	. 2 ⊢ ((𝜓 ⊼ 𝜑) ↔ (𝜓 → ¬ 𝜑))
4	1, 2, 3	3bitr4i 291	1 ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ⊼ wnan 1439

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 196  df-an 385  df-nan 1440

This theorem is referenced by: (None)