Theorem ex-natded9.20-2 27275

Description: A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 27274. (Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ex-natded9.20.1	|- ( ph -> ( ps /\ ( ch \/ th ) ) )

Assertion

Ref	Expression
ex-natded9.20-2	|- ( ph -> ( ( ps /\ ch ) \/ ( ps /\ th ) ) )

Proof of Theorem ex-natded9.20-2

Step	Hyp	Ref	Expression
1		ex-natded9.20.1	. . . . 5 |- ( ph -> ( ps /\ ( ch \/ th ) ) )
2	1	simpld 475	. . . 4 |- ( ph -> ps )
3	2	anim1i 592	. . 3 |- ( ( ph /\ ch ) -> ( ps /\ ch ) )
4	3	orcd 407	. 2 |- ( ( ph /\ ch ) -> ( ( ps /\ ch ) \/ ( ps /\ th ) ) )
5	2	anim1i 592	. . 3 |- ( ( ph /\ th ) -> ( ps /\ th ) )
6	5	olcd 408	. 2 |- ( ( ph /\ th ) -> ( ( ps /\ ch ) \/ ( ps /\ th ) ) )
7	1	simprd 479	. 2 |- ( ph -> ( ch \/ th ) )
8	4, 6, 7	mpjaodan 827	1 |- ( ph -> ( ( ps /\ ch ) \/ ( ps /\ th ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ 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-or 385  df-an 386

This theorem is referenced by: (None)