Theorem nanorxor 38504

Description: 'nand' is equivalent to the equivalence of inclusive and exclusive or. (Contributed by Steve Rodriguez, 28-Feb-2020.)

Assertion

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

Proof of Theorem nanorxor

Step	Hyp	Ref	Expression
1		df-nan 1448	. 2 |- ( ( ph -/\ ps ) <-> -. ( ph /\ ps ) )
2		xor2 1470	. . . 4 |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
3	2	rbaibr 946	. . 3 |- ( -. ( ph /\ ps ) -> ( ( ph \/ ps ) <-> ( ph \/_ ps ) ) )
4	2	bibi2i 327	. . . 4 |- ( ( ( ph \/ ps ) <-> ( ph \/_ ps ) ) <-> ( ( ph \/ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) )
5		pm4.71 662	. . . . 5 |- ( ( ( ph \/ ps ) -> -. ( ph /\ ps ) ) <-> ( ( ph \/ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) )
6		simpl 473	. . . . . . . 8 |- ( ( ph /\ ps ) -> ph )
7	6	orcd 407	. . . . . . 7 |- ( ( ph /\ ps ) -> ( ph \/ ps ) )
8	7	con3i 150	. . . . . 6 |- ( -. ( ph \/ ps ) -> -. ( ph /\ ps ) )
9		id 22	. . . . . 6 |- ( -. ( ph /\ ps ) -> -. ( ph /\ ps ) )
10	8, 9	ja 173	. . . . 5 |- ( ( ( ph \/ ps ) -> -. ( ph /\ ps ) ) -> -. ( ph /\ ps ) )
11	5, 10	sylbir 225	. . . 4 |- ( ( ( ph \/ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) -> -. ( ph /\ ps ) )
12	4, 11	sylbi 207	. . 3 |- ( ( ( ph \/ ps ) <-> ( ph \/_ ps ) ) -> -. ( ph /\ ps ) )
13	3, 12	impbii 199	. 2 |- ( -. ( ph /\ ps ) <-> ( ( ph \/ ps ) <-> ( ph \/_ ps ) ) )
14	1, 13	bitri 264	1 |- ( ( ph -/\ ps ) <-> ( ( ph \/ ps ) <-> ( ph \/_ ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   -/\ wnan 1447   \/_ 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-nan 1448  df-xor 1465

This theorem is referenced by:  undisjrab  38505