Theorem xorcom 1543

Description: The connector ⊻ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)

Assertion

Ref	Expression
xorcom	⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑))

Proof of Theorem xorcom

Step	Hyp	Ref	Expression
1		df-xor 1541	. . 3 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
2		bicom 225	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑))
3	1, 2	xchbinx 337	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜓 ↔ 𝜑))
4		df-xor 1541	. 2 ⊢ ((𝜓 ⊻ 𝜑) ↔ ¬ (𝜓 ↔ 𝜑))
5	3, 4	bitr4i 281	1 ⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ⊻ wxo 1540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-xor 1541

This theorem is used by:  xorneg1  1551  falxortru  1616  hadcomb  1629  cadcoma  1641  oneptri  44012