Theorem nabctnabc 41098

Description: not ( a -> ( b /\ c ) ) we can show: not a implies ( b /\ c ). (Contributed by Jarvin Udandy, 7-Sep-2020.)

Hypothesis

Ref	Expression
nabctnabc.1	|- -. ( ph -> ( ps /\ ch ) )

Assertion

Ref	Expression
nabctnabc	|- ( -. ph -> ( ps /\ ch ) )

Proof of Theorem nabctnabc

Step	Hyp	Ref	Expression
1		nabctnabc.1	. . . . . . . 8 |- -. ( ph -> ( ps /\ ch ) )
2		pm4.61 442	. . . . . . . . 9 |- ( -. ( ph -> ( ps /\ ch ) ) <-> ( ph /\ -. ( ps /\ ch ) ) )
3	2	biimpi 206	. . . . . . . 8 |- ( -. ( ph -> ( ps /\ ch ) ) -> ( ph /\ -. ( ps /\ ch ) ) )
4	1, 3	ax-mp 5	. . . . . . 7 |- ( ph /\ -. ( ps /\ ch ) )
5	4	simpli 474	. . . . . 6 |- ph
6	4	simpri 478	. . . . . 6 |- -. ( ps /\ ch )
7	5, 6	2th 254	. . . . 5 |- ( ph <-> -. ( ps /\ ch ) )
8		bicom 212	. . . . . 6 |- ( ( ph <-> -. ( ps /\ ch ) ) <-> ( -. ( ps /\ ch ) <-> ph ) )
9	8	biimpi 206	. . . . 5 |- ( ( ph <-> -. ( ps /\ ch ) ) -> ( -. ( ps /\ ch ) <-> ph ) )
10	7, 9	ax-mp 5	. . . 4 |- ( -. ( ps /\ ch ) <-> ph )
11	10	biimpi 206	. . 3 |- ( -. ( ps /\ ch ) -> ph )
12	11	con3i 150	. 2 |- ( -. ph -> -. -. ( ps /\ ch ) )
13	12	notnotrd 128	1 |- ( -. ph -> ( ps /\ ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384

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

This theorem is referenced by: (None)