Theorem xor 500

Description: Two ways to express "exclusive or". Theorem *5.22 of [WhiteheadRussell] p. 124.

Assertion

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

Proof of Theorem xor

Step	Hyp	Ref	Expression
1		dfbi 499	. 2 |- ((-. ph <-> ps) <-> ((-. ph /\ ps) \/ (-. -. ph /\ -. ps)))
2		nbbn 498	. 2 |- ((-. ph <-> ps) <-> -. (ph <-> ps))
3		ancom 333	. . . 4 |- ((ps /\ -. ph) <-> (-. ph /\ ps))
4		pm4.13 142	. . . . 5 |- (ph <-> -. -. ph)
5	4	anbi1i 368	. . . 4 |- ((ph /\ -. ps) <-> (-. -. ph /\ -. ps))
6	3, 5	orbi12i 216	. . 3 |- (((ps /\ -. ph) \/ (ph /\ -. ps)) <-> ((-. ph /\ ps) \/ (-. -. ph /\ -. ps)))
7		orcom 209	. . 3 |- (((ps /\ -. ph) \/ (ph /\ -. ps)) <-> ((ph /\ -. ps) \/ (ps /\ -. ph)))
8	6, 7	bitr3 153	. 2 |- (((-. ph /\ ps) \/ (-. -. ph /\ -. ps)) <-> ((ph /\ -. ps) \/ (ps /\ -. ph)))
9	1, 2, 8	3bitr3 156	1 |- (-. (ph <-> ps) <-> ((ph /\ -. ps) \/ (ps /\ -. ph)))

Syntax hints:  -. wn 1   <-> wb 127   \/ wo 195   /\ wa 196

This theorem is referenced by:  biass 511

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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