Theorem dfxor5

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

	Ref	Expression
Assertion	dfxor5	⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ¬ ( ( 𝜑 → ¬ 𝜓 ) → ¬ ( ¬ 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfxor4	⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ¬ ( ( ¬ 𝜑 → 𝜓 ) → ¬ ( 𝜑 → ¬ 𝜓 ) ) )
2		con2b	⊢ ( ( ( ¬ 𝜑 → 𝜓 ) → ¬ ( 𝜑 → ¬ 𝜓 ) ) ↔ ( ( 𝜑 → ¬ 𝜓 ) → ¬ ( ¬ 𝜑 → 𝜓 ) ) )
3	1 2	xchbinx	⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ¬ ( ( 𝜑 → ¬ 𝜓 ) → ¬ ( ¬ 𝜑 → 𝜓 ) ) )