Theorem nanorxor

Description: 'nand' is equivalent to the equivalence of inclusive and exclusive or. (Contributed by Steve Rodriguez, 28-Feb-2020)

	Ref	Expression
Assertion	nanorxor	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ⊻ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-nan	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ ¬ ( 𝜑 ∧ 𝜓 ) )
2		xor2	⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜓 ) ) )
3	2	rbaibr	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ⊻ 𝜓 ) ) )
4	2	bibi2i	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ⊻ 𝜓 ) ) ↔ ( ( 𝜑 ∨ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜓 ) ) ) )
5		pm4.71	⊢ ( ( ( 𝜑 ∨ 𝜓 ) → ¬ ( 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝜑 ∨ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜓 ) ) ) )
6		simpl	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 )
7	6	orcd	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∨ 𝜓 ) )
8	7	con3i	⊢ ( ¬ ( 𝜑 ∨ 𝜓 ) → ¬ ( 𝜑 ∧ 𝜓 ) )
9		id	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) → ¬ ( 𝜑 ∧ 𝜓 ) )
10	8 9	ja	⊢ ( ( ( 𝜑 ∨ 𝜓 ) → ¬ ( 𝜑 ∧ 𝜓 ) ) → ¬ ( 𝜑 ∧ 𝜓 ) )
11	5 10	sylbir	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜓 ) ) ) → ¬ ( 𝜑 ∧ 𝜓 ) )
12	4 11	sylbi	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ⊻ 𝜓 ) ) → ¬ ( 𝜑 ∧ 𝜓 ) )
13	3 12	impbii	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ⊻ 𝜓 ) ) )
14	1 13	bitri	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ⊻ 𝜓 ) ) )