Theorem ex-natded5.13-2 27273

Description: A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 27272. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ex-natded5.13.1	|- ( ph -> ( ps \/ ch ) )
ex-natded5.13.2	|- ( ph -> ( ps -> th ) )
ex-natded5.13.3	|- ( ph -> ( -. ta -> -. ch ) )

Assertion

Ref	Expression
ex-natded5.13-2	|- ( ph -> ( th \/ ta ) )

Proof of Theorem ex-natded5.13-2

Step	Hyp	Ref	Expression
1		ex-natded5.13.1	. 2 |- ( ph -> ( ps \/ ch ) )
2		ex-natded5.13.2	. . 3 |- ( ph -> ( ps -> th ) )
3		ex-natded5.13.3	. . . 4 |- ( ph -> ( -. ta -> -. ch ) )
4	3	con4d 114	. . 3 |- ( ph -> ( ch -> ta ) )
5	2, 4	orim12d 883	. 2 |- ( ph -> ( ( ps \/ ch ) -> ( th \/ ta ) ) )
6	1, 5	mpd 15	1 |- ( ph -> ( th \/ ta ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383

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-or 385  df-an 386

This theorem is referenced by: (None)