Theorem waj-ax 32413

Description: A single axiom for propositional calculus offered by Wajsberg. (Contributed by Anthony Hart, 13-Aug-2011.)

Assertion

Ref	Expression
waj-ax	|- ( ( ph -/\ ( ps -/\ ch ) ) -/\ ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) -/\ ( ph -/\ ( ph -/\ ps ) ) ) )

Proof of Theorem waj-ax

Step	Hyp	Ref	Expression
1		nannan 1451	. . 3 |- ( ( ph -/\ ( ps -/\ ch ) ) <-> ( ph -> ( ps /\ ch ) ) )
2		simpr 477	. . . . . . . . 9 |- ( ( ps /\ ch ) -> ch )
3	2	imim2i 16	. . . . . . . 8 |- ( ( ph -> ( ps /\ ch ) ) -> ( ph -> ch ) )
4		pm2.27 42	. . . . . . . . . 10 |- ( ph -> ( ( ph -> ch ) -> ch ) )
5	4	anim2d 589	. . . . . . . . 9 |- ( ph -> ( ( th /\ ( ph -> ch ) ) -> ( th /\ ch ) ) )
6	5	expdimp 453	. . . . . . . 8 |- ( ( ph /\ th ) -> ( ( ph -> ch ) -> ( th /\ ch ) ) )
7	3, 6	syl5com 31	. . . . . . 7 |- ( ( ph -> ( ps /\ ch ) ) -> ( ( ph /\ th ) -> ( th /\ ch ) ) )
8	7	con3d 148	. . . . . 6 |- ( ( ph -> ( ps /\ ch ) ) -> ( -. ( th /\ ch ) -> -. ( ph /\ th ) ) )
9		df-nan 1448	. . . . . 6 |- ( ( th -/\ ch ) <-> -. ( th /\ ch ) )
10		df-nan 1448	. . . . . 6 |- ( ( ph -/\ th ) <-> -. ( ph /\ th ) )
11	8, 9, 10	3imtr4g 285	. . . . 5 |- ( ( ph -> ( ps /\ ch ) ) -> ( ( th -/\ ch ) -> ( ph -/\ th ) ) )
12		nanim 1452	. . . . 5 |- ( ( ( th -/\ ch ) -> ( ph -/\ th ) ) <-> ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) )
13	11, 12	sylib 208	. . . 4 |- ( ( ph -> ( ps /\ ch ) ) -> ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) )
14		pm3.21 464	. . . . . . . 8 |- ( ps -> ( ph -> ( ph /\ ps ) ) )
15	14	adantr 481	. . . . . . 7 |- ( ( ps /\ ch ) -> ( ph -> ( ph /\ ps ) ) )
16	15	com12 32	. . . . . 6 |- ( ph -> ( ( ps /\ ch ) -> ( ph /\ ps ) ) )
17	16	a2i 14	. . . . 5 |- ( ( ph -> ( ps /\ ch ) ) -> ( ph -> ( ph /\ ps ) ) )
18		nannan 1451	. . . . 5 |- ( ( ph -/\ ( ph -/\ ps ) ) <-> ( ph -> ( ph /\ ps ) ) )
19	17, 18	sylibr 224	. . . 4 |- ( ( ph -> ( ps /\ ch ) ) -> ( ph -/\ ( ph -/\ ps ) ) )
20	13, 19	jca 554	. . 3 |- ( ( ph -> ( ps /\ ch ) ) -> ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) /\ ( ph -/\ ( ph -/\ ps ) ) ) )
21	1, 20	sylbi 207	. 2 |- ( ( ph -/\ ( ps -/\ ch ) ) -> ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) /\ ( ph -/\ ( ph -/\ ps ) ) ) )
22		nannan 1451	. 2 |- ( ( ( ph -/\ ( ps -/\ ch ) ) -/\ ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) -/\ ( ph -/\ ( ph -/\ ps ) ) ) ) <-> ( ( ph -/\ ( ps -/\ ch ) ) -> ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) /\ ( ph -/\ ( ph -/\ ps ) ) ) ) )
23	21, 22	mpbir 221	1 |- ( ( ph -/\ ( ps -/\ ch ) ) -/\ ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) -/\ ( ph -/\ ( ph -/\ ps ) ) ) )

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

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

This theorem is referenced by: (None)