Theorem xorcom 1388

Description: ⊻ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.)

Assertion

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

Proof of Theorem xorcom

Step	Hyp	Ref	Expression
1		orcom 728	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
2		ancom 266	. . . 4 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
3	2	notbii 668	. . 3 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ ¬ (𝜓 ∧ 𝜑))
4	1, 3	anbi12i 460	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ ((𝜓 ∨ 𝜑) ∧ ¬ (𝜓 ∧ 𝜑)))
5		df-xor 1376	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
6		df-xor 1376	. 2 ⊢ ((𝜓 ⊻ 𝜑) ↔ ((𝜓 ∨ 𝜑) ∧ ¬ (𝜓 ∧ 𝜑)))
7	4, 5, 6	3bitr4i 212	1 ⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑))

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 708   ⊻ wxo 1375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-xor 1376

This theorem is referenced by:  rpnegap  9682