Theorem xorbin 1384

Description: A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.)

Assertion

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

Proof of Theorem xorbin

Step	Hyp	Ref	Expression
1		df-xor 1376	. . 3 |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
2		imnan 690	. . . . 5 |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
3	2	biimpri 133	. . . 4 |- ( -. ( ph /\ ps ) -> ( ph -> -. ps ) )
4	3	adantl 277	. . 3 |- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) -> ( ph -> -. ps ) )
5	1, 4	sylbi 121	. 2 |- ( ( ph \/_ ps ) -> ( ph -> -. ps ) )
6		pm2.53 722	. . . . 5 |- ( ( ps \/ ph ) -> ( -. ps -> ph ) )
7	6	orcoms 730	. . . 4 |- ( ( ph \/ ps ) -> ( -. ps -> ph ) )
8	7	adantr 276	. . 3 |- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) -> ( -. ps -> ph ) )
9	1, 8	sylbi 121	. 2 |- ( ( ph \/_ ps ) -> ( -. ps -> ph ) )
10	5, 9	impbid 129	1 |- ( ( ph \/_ ps ) -> ( ph <-> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   \/_ wxo 1375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-xor 1376

This theorem is referenced by:  xornbi  1386  zeo4  11877  odd2np1  11880