Theorem mpan10 474

Description: Modus ponens mixed with several conjunctions. (Contributed by Jim Kingdon, 7-Jan-2018.)

Assertion

Ref	Expression
mpan10	|- ( ( ( ( ph -> ps ) /\ ch ) /\ ph ) -> ( ps /\ ch ) )

Proof of Theorem mpan10

Step	Hyp	Ref	Expression
1		ancom 266	. . . 4 |- ( ( ch /\ ph ) <-> ( ph /\ ch ) )
2	1	anbi2i 457	. . 3 |- ( ( ( ph -> ps ) /\ ( ch /\ ph ) ) <-> ( ( ph -> ps ) /\ ( ph /\ ch ) ) )
3		anass 401	. . 3 |- ( ( ( ( ph -> ps ) /\ ch ) /\ ph ) <-> ( ( ph -> ps ) /\ ( ch /\ ph ) ) )
4		anass 401	. . 3 |- ( ( ( ( ph -> ps ) /\ ph ) /\ ch ) <-> ( ( ph -> ps ) /\ ( ph /\ ch ) ) )
5	2, 3, 4	3bitr4i 212	. 2 |- ( ( ( ( ph -> ps ) /\ ch ) /\ ph ) <-> ( ( ( ph -> ps ) /\ ph ) /\ ch ) )
6		id 19	. . . 4 |- ( ( ph -> ps ) -> ( ph -> ps ) )
7	6	imp 124	. . 3 |- ( ( ( ph -> ps ) /\ ph ) -> ps )
8	7	anim1i 340	. 2 |- ( ( ( ( ph -> ps ) /\ ph ) /\ ch ) -> ( ps /\ ch ) )
9	5, 8	sylbi 121	1 |- ( ( ( ( ph -> ps ) /\ ch ) /\ ph ) -> ( ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)