Theorem xor 671

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		dfbi3 670	. 2 |- ((-. ph <-> ps) <-> ((-. ph /\ ps) \/ (-. -. ph /\ -. ps)))
2		nbbn 661	. 2 |- ((-. ph <-> ps) <-> -. (ph <-> ps))
3		ancom 435	. . . 4 |- ((ps /\ -. ph) <-> (-. ph /\ ps))
4		pm4.13 161	. . . . 5 |- (ph <-> -. -. ph)
5	4	anbi1i 481	. . . 4 |- ((ph /\ -. ps) <-> (-. -. ph /\ -. ps))
6	3, 5	orbi12i 257	. . 3 |- (((ps /\ -. ph) \/ (ph /\ -. ps)) <-> ((-. ph /\ ps) \/ (-. -. ph /\ -. ps)))
7		orcom 246	. . 3 |- (((ps /\ -. ph) \/ (ph /\ -. ps)) <-> ((ph /\ -. ps) \/ (ps /\ -. ph)))
8	6, 7	bitr3 175	. 2 |- (((-. ph /\ ps) \/ (-. -. ph /\ -. ps)) <-> ((ph /\ -. ps) \/ (ps /\ -. ph)))
9	1, 2, 8	3bitr3 181	1 |- (-. (ph <-> ps) <-> ((ph /\ -. ps) \/ (ps /\ -. ph)))

Syntax hints:  -. wn 2   <-> wb 146   \/ wo 222   /\ wa 223

This theorem is referenced by:  pm5.24 672  xor2 673

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225

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