Theorem norcom 1558

Description: The connector ⊽ is commutative. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 23-Apr-2024.)

Assertion

Ref	Expression
norcom	⊢ ((𝜑 ⊽ 𝜓) ↔ (𝜓 ⊽ 𝜑))

Proof of Theorem norcom

Step	Hyp	Ref	Expression
1		df-nor 1557	. . 3 ⊢ ((𝜑 ⊽ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))
2		orcom 883	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
3	1, 2	xchbinx 337	. 2 ⊢ ((𝜑 ⊽ 𝜓) ↔ ¬ (𝜓 ∨ 𝜑))
4		df-nor 1557	. 2 ⊢ ((𝜓 ⊽ 𝜑) ↔ ¬ (𝜓 ∨ 𝜑))
5	3, 4	bitr4i 281	1 ⊢ ((𝜑 ⊽ 𝜓) ↔ (𝜓 ⊽ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∨ wo 860   ⊽ wnor 1556

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 210  df-or 861  df-nor 1557

This theorem is referenced by:  norass  1565  falnortru  1619