Theorem contrd 33899

Description: A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)

Hypotheses

Ref	Expression
contrd.1	|- ( ph -> ( -. ps -> ch ) )
contrd.2	|- ( ph -> ( -. ps -> -. ch ) )

Assertion

Ref	Expression
contrd	|- ( ph -> ps )

Proof of Theorem contrd

Step	Hyp	Ref	Expression
1		contrd.1	. . 3 |- ( ph -> ( -. ps -> ch ) )
2		contrd.2	. . 3 |- ( ph -> ( -. ps -> -. ch ) )
3	1, 2	jcad 555	. 2 |- ( ph -> ( -. ps -> ( ch /\ -. ch ) ) )
4		pm2.24 121	. . . . 5 |- ( ch -> ( -. ch -> ps ) )
5	4	imp 445	. . . 4 |- ( ( ch /\ -. ch ) -> ps )
6	5	imim2i 16	. . 3 |- ( ( -. ps -> ( ch /\ -. ch ) ) -> ( -. ps -> ps ) )
7	6	pm2.18d 124	. 2 |- ( ( -. ps -> ( ch /\ -. ch ) ) -> ps )
8	3, 7	syl 17	1 |- ( ph -> ps )

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:  mpt2bi123f  33971  mptbi12f  33975  ac6s6  33980