Theorem nandsym1 32421

Description: A symmetry with -/\.

See negsym1 32416 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

Assertion

Ref	Expression
nandsym1	|- ( ( ps -/\ ( ps -/\ F. ) ) -> ( ps -/\ ph ) )

Proof of Theorem nandsym1

Step	Hyp	Ref	Expression
1		df-nan 1448	. . . . 5 |- ( ( ps -/\ ( ps -/\ F. ) ) <-> -. ( ps /\ ( ps -/\ F. ) ) )
2	1	biimpi 206	. . . 4 |- ( ( ps -/\ ( ps -/\ F. ) ) -> -. ( ps /\ ( ps -/\ F. ) ) )
3		df-nan 1448	. . . . 5 |- ( ( ps -/\ F. ) <-> -. ( ps /\ F. ) )
4	3	anbi2i 730	. . . 4 |- ( ( ps /\ ( ps -/\ F. ) ) <-> ( ps /\ -. ( ps /\ F. ) ) )
5	2, 4	sylnib 318	. . 3 |- ( ( ps -/\ ( ps -/\ F. ) ) -> -. ( ps /\ -. ( ps /\ F. ) ) )
6		simpl 473	. . . 4 |- ( ( ps /\ ph ) -> ps )
7		fal 1490	. . . . 5 |- -. F.
8	7	intnan 960	. . . 4 |- -. ( ps /\ F. )
9	6, 8	jctir 561	. . 3 |- ( ( ps /\ ph ) -> ( ps /\ -. ( ps /\ F. ) ) )
10	5, 9	nsyl 135	. 2 |- ( ( ps -/\ ( ps -/\ F. ) ) -> -. ( ps /\ ph ) )
11		df-nan 1448	. 2 |- ( ( ps -/\ ph ) <-> -. ( ps /\ ph ) )
12	10, 11	sylibr 224	1 |- ( ( ps -/\ ( ps -/\ F. ) ) -> ( ps -/\ ph ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   -/\ wnan 1447   F. wfal 1488

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-an 386  df-nan 1448  df-tru 1486  df-fal 1489

This theorem is referenced by: (None)