Metamath Proof Explorer

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	\|- ( ( ph \/_ ps ) <-> -. ( ( ph -> -. ps ) -> -. ( -. ph -> ps ) ) )

Step	Hyp	Ref	Expression
1		dfxor4	\|- ( ( ph \/_ ps ) <-> -. ( ( -. ph -> ps ) -> -. ( ph -> -. ps ) ) )
2		con2b	\|- ( ( ( -. ph -> ps ) -> -. ( ph -> -. ps ) ) <-> ( ( ph -> -. ps ) -> -. ( -. ph -> ps ) ) )
3	1 2	xchbinx	\|- ( ( ph \/_ ps ) <-> -. ( ( ph -> -. ps ) -> -. ( -. ph -> ps ) ) )

Step

Hyp

Ref

Expression

 |-  ( ( ph \/_ ps ) <-> -. ( ( -. ph -> ps ) -> -. ( ph -> -. ps ) ) )

 |-  ( ( ( -. ph -> ps ) -> -. ( ph -> -. ps ) ) <-> ( ( ph -> -. ps ) -> -. ( -. ph -> ps ) ) )

1 2

 |-  ( ( ph \/_ ps ) <-> -. ( ( ph -> -. ps ) -> -. ( -. ph -> ps ) ) )