Theorem xorcom 1279

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 647	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
2		ancom 253	. . . 4 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
3	2	notbii 594	. . 3 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ ¬ (𝜓 ∧ 𝜑))
4	1, 3	anbi12i 433	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ ((𝜓 ∨ 𝜑) ∧ ¬ (𝜓 ∧ 𝜑)))
5		df-xor 1267	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
6		df-xor 1267	. 2 ⊢ ((𝜓 ⊻ 𝜑) ↔ ((𝜓 ∨ 𝜑) ∧ ¬ (𝜓 ∧ 𝜑)))
7	4, 5, 6	3bitr4i 201	1 ⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑))

Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98   ∨ wo 629   ⊻ wxo 1266

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-xor 1267

This theorem is referenced by:  rpnegap  8615