Theorem dfxor4 38058

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

Assertion

Ref	Expression
dfxor4	|- ( ( ph \/_ ps ) <-> -. ( ( -. ph -> ps ) -> -. ( ph -> -. ps ) ) )

Proof of Theorem dfxor4

Step	Hyp	Ref	Expression
1		xor2 1470	. 2 |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
2		df-or 385	. . 3 |- ( ( ph \/ ps ) <-> ( -. ph -> ps ) )
3		imnan 438	. . . 4 |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
4	3	bicomi 214	. . 3 |- ( -. ( ph /\ ps ) <-> ( ph -> -. ps ) )
5	2, 4	anbi12i 733	. 2 |- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ( -. ph -> ps ) /\ ( ph -> -. ps ) ) )
6		df-an 386	. 2 |- ( ( ( -. ph -> ps ) /\ ( ph -> -. ps ) ) <-> -. ( ( -. ph -> ps ) -> -. ( ph -> -. ps ) ) )
7	1, 5, 6	3bitri 286	1 |- ( ( ph \/_ ps ) <-> -. ( ( -. ph -> ps ) -> -. ( ph -> -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/_ wxo 1464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-xor 1465

This theorem is referenced by:  dfxor5  38059