Theorem ex-natded5.2-2 27262

Description: A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 27261 and ex-natded5.2i 27263. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ex-natded5.2.1	|- ( ph -> ( ( ps /\ ch ) -> th ) )
ex-natded5.2.2	|- ( ph -> ( ch -> ps ) )
ex-natded5.2.3	|- ( ph -> ch )

Assertion

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

Proof of Theorem ex-natded5.2-2

Step	Hyp	Ref	Expression
1		ex-natded5.2.3	. . 3 |- ( ph -> ch )
2		ex-natded5.2.2	. . 3 |- ( ph -> ( ch -> ps ) )
3	1, 2	mpd 15	. 2 |- ( ph -> ps )
4		ex-natded5.2.1	. 2 |- ( ph -> ( ( ps /\ ch ) -> th ) )
5	3, 1, 4	mp2and 715	1 |- ( ph -> th )

Syntax hints:   -> 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)