Theorem nancom 1442

Description: The 'nand' operator commutes. (Contributed by Mario Carneiro, 9-May-2015.) (Proof shortened by Wolf Lammen, 7-Mar-2020.)

Assertion

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

Proof of Theorem nancom

Step	Hyp	Ref	Expression
1		df-nan 1440	. . 3 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
2		ancom 465	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
3	1, 2	xchbinx 323	. 2 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜓 ∧ 𝜑))
4		df-nan 1440	. 2 ⊢ ((𝜓 ⊼ 𝜑) ↔ ¬ (𝜓 ∧ 𝜑))
5	3, 4	bitr4i 266	1 ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   ⊼ 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:  nanbi2  1448  falnantru  1517  rp-fakenanass  36879