Theorem ex-natded5.8-2 27271

Description: A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 27270. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ex-natded5.8.1	|- ( ph -> ( ( ps /\ ch ) -> -. th ) )
ex-natded5.8.2	|- ( ph -> ( ta -> th ) )
ex-natded5.8.3	|- ( ph -> ch )
ex-natded5.8.4	|- ( ph -> ta )

Assertion

Ref	Expression
ex-natded5.8-2	|- ( ph -> -. ps )

Proof of Theorem ex-natded5.8-2

Step	Hyp	Ref	Expression
1		ex-natded5.8.4	. . 3 |- ( ph -> ta )
2		ex-natded5.8.2	. . 3 |- ( ph -> ( ta -> th ) )
3	1, 2	mpd 15	. 2 |- ( ph -> th )
4		ex-natded5.8.3	. . 3 |- ( ph -> ch )
5		ex-natded5.8.1	. . 3 |- ( ph -> ( ( ps /\ ch ) -> -. th ) )
6	4, 5	mpan2d 710	. 2 |- ( ph -> ( ps -> -. th ) )
7	3, 6	mt2d 131	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: (None)