Theorem xor 861

Description: Two ways to express "exclusive or." Theorem *5.22 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 22-Jan-2013.)

Assertion

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

Proof of Theorem xor

Step	Hyp	Ref	Expression
1		iman 413	. . . 4 |- ((ph -> ps) <-> -. (ph /\ -. ps))
2		iman 413	. . . 4 |- ((ps -> ph) <-> -. (ps /\ -. ph))
3	1, 2	anbi12i 678	. . 3 |- (((ph -> ps) /\ (ps -> ph)) <-> (-. (ph /\ -. ps) /\ -. (ps /\ -. ph)))
4		dfbi2 609	. . 3 |- ((ph <-> ps) <-> ((ph -> ps) /\ (ps -> ph)))
5		ioran 476	. . 3 |- (-. ((ph /\ -. ps) \/ (ps /\ -. ph)) <-> (-. (ph /\ -. ps) /\ -. (ps /\ -. ph)))
6	3, 4, 5	3bitr4ri 269	. 2 |- (-. ((ph /\ -. ps) \/ (ps /\ -. ph)) <-> (ph <-> ps))
7	6	con1bii 321	1 |- (-. (ph <-> ps) <-> ((ph /\ -. ps) \/ (ps /\ -. ph)))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358

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

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is referenced by:  dfbi3  863  pm5.24  864  4exmid  905  excxor  1309  elsymdif  3223  symdif2  3520