Theorem exmid2 33901

Description: An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.)

Hypotheses

Ref	Expression
exmid2.1	|- ( ( ps /\ ph ) -> ch )
exmid2.2	|- ( ( -. ps /\ et ) -> ch )

Assertion

Ref	Expression
exmid2	|- ( ( ph /\ et ) -> ch )

Proof of Theorem exmid2

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 |- ( ( ph /\ et ) -> ph )
2	1	anim2i 593	. . . 4 |- ( ( ps /\ ( ph /\ et ) ) -> ( ps /\ ph ) )
3	2	ancoms 469	. . 3 |- ( ( ( ph /\ et ) /\ ps ) -> ( ps /\ ph ) )
4		exmid2.1	. . 3 |- ( ( ps /\ ph ) -> ch )
5	3, 4	syl 17	. 2 |- ( ( ( ph /\ et ) /\ ps ) -> ch )
6		simpr 477	. . . . 5 |- ( ( ph /\ et ) -> et )
7	6	anim2i 593	. . . 4 |- ( ( -. ps /\ ( ph /\ et ) ) -> ( -. ps /\ et ) )
8	7	ancoms 469	. . 3 |- ( ( ( ph /\ et ) /\ -. ps ) -> ( -. ps /\ et ) )
9		exmid2.2	. . 3 |- ( ( -. ps /\ et ) -> ch )
10	8, 9	syl 17	. 2 |- ( ( ( ph /\ et ) /\ -. ps ) -> ch )
11	5, 10	pm2.61dan 832	1 |- ( ( ph /\ et ) -> ch )

Syntax hints:   -. wn 3   -> wi 4   /\ 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)