Theorem dfxor4

Description: Express exclusive-or in terms of implication and negation. Statement in Frege1879 p. 12. (Contributed by RP, 14-Apr-2020)

	Ref	Expression
Assertion	dfxor4	⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ¬ ( ( ¬ 𝜑 → 𝜓 ) → ¬ ( 𝜑 → ¬ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		xor2	⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜓 ) ) )
2		df-or	⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ 𝜑 → 𝜓 ) )
3		imnan	⊢ ( ( 𝜑 → ¬ 𝜓 ) ↔ ¬ ( 𝜑 ∧ 𝜓 ) )
4	3	bicomi	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 → ¬ 𝜓 ) )
5	2 4	anbi12i	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜓 ) ) ↔ ( ( ¬ 𝜑 → 𝜓 ) ∧ ( 𝜑 → ¬ 𝜓 ) ) )
6		df-an	⊢ ( ( ( ¬ 𝜑 → 𝜓 ) ∧ ( 𝜑 → ¬ 𝜓 ) ) ↔ ¬ ( ( ¬ 𝜑 → 𝜓 ) → ¬ ( 𝜑 → ¬ 𝜓 ) ) )
7	1 5 6	3bitri	⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ¬ ( ( ¬ 𝜑 → 𝜓 ) → ¬ ( 𝜑 → ¬ 𝜓 ) ) )