Theorem xorcom 1276

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 646	. . 3 ⊢ ((φ ∨ ψ) ↔ (ψ ∨ φ))
2		ancom 253	. . . 4 ⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ))
3	2	notbii 593	. . 3 ⊢ (¬ (φ ∧ ψ) ↔ ¬ (ψ ∧ φ))
4	1, 3	anbi12i 433	. 2 ⊢ (((φ ∨ ψ) ∧ ¬ (φ ∧ ψ)) ↔ ((ψ ∨ φ) ∧ ¬ (ψ ∧ φ)))
5		df-xor 1266	. 2 ⊢ ((φ ⊻ ψ) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ)))
6		df-xor 1266	. 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 628   ⊻ wxo 1265

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 629

This theorem depends on definitions:  df-bi 110  df-xor 1266

This theorem is referenced by:  rpnegap  8350