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	$$⊢\left(\phi ⊻\psi \right)\leftrightarrow \neg \left(\left(\phi \to \neg \psi \right)\to \neg \left(\neg \phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfxor4	$$⊢\left(\phi ⊻\psi \right)\leftrightarrow \neg \left(\left(\neg \phi \to \psi \right)\to \neg \left(\phi \to \neg \psi \right)\right)$$
2		con2b	$$⊢\left(\left(\neg \phi \to \psi \right)\to \neg \left(\phi \to \neg \psi \right)\right)\leftrightarrow \left(\left(\phi \to \neg \psi \right)\to \neg \left(\neg \phi \to \psi \right)\right)$$
3	1 2	xchbinx	$$⊢\left(\phi ⊻\psi \right)\leftrightarrow \neg \left(\left(\phi \to \neg \psi \right)\to \neg \left(\neg \phi \to \psi \right)\right)$$