Theorem uun2221p1 39041

Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
uun2221p1.1	|- ( ( ph /\ ( ps /\ ph ) /\ ph ) -> ch )

Assertion

Ref	Expression
uun2221p1	|- ( ( ps /\ ph ) -> ch )

Proof of Theorem uun2221p1

Step	Hyp	Ref	Expression
1		uun2221p1.1	. . 3 |- ( ( ph /\ ( ps /\ ph ) /\ ph ) -> ch )
2		3anrot 1043	. . . 4 |- ( ( ph /\ ph /\ ( ps /\ ph ) ) <-> ( ph /\ ( ps /\ ph ) /\ ph ) )
3	2	imbi1i 339	. . 3 |- ( ( ( ph /\ ph /\ ( ps /\ ph ) ) -> ch ) <-> ( ( ph /\ ( ps /\ ph ) /\ ph ) -> ch ) )
4	1, 3	mpbir 221	. 2 |- ( ( ph /\ ph /\ ( ps /\ ph ) ) -> ch )
5		3anass 1042	. . . . . 6 |- ( ( ph /\ ph /\ ( ps /\ ph ) ) <-> ( ph /\ ( ph /\ ( ps /\ ph ) ) ) )
6		anabs5 851	. . . . . 6 |- ( ( ph /\ ( ph /\ ( ps /\ ph ) ) ) <-> ( ph /\ ( ps /\ ph ) ) )
7	5, 6	bitri 264	. . . . 5 |- ( ( ph /\ ph /\ ( ps /\ ph ) ) <-> ( ph /\ ( ps /\ ph ) ) )
8		ancom 466	. . . . . 6 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
9	8	anbi2i 730	. . . . 5 |- ( ( ph /\ ( ph /\ ps ) ) <-> ( ph /\ ( ps /\ ph ) ) )
10	7, 9	bitr4i 267	. . . 4 |- ( ( ph /\ ph /\ ( ps /\ ph ) ) <-> ( ph /\ ( ph /\ ps ) ) )
11		anabs5 851	. . . . 5 |- ( ( ph /\ ( ph /\ ps ) ) <-> ( ph /\ ps ) )
12	11, 8	bitri 264	. . . 4 |- ( ( ph /\ ( ph /\ ps ) ) <-> ( ps /\ ph ) )
13	10, 12	bitri 264	. . 3 |- ( ( ph /\ ph /\ ( ps /\ ph ) ) <-> ( ps /\ ph ) )
14	13	imbi1i 339	. 2 |- ( ( ( ph /\ ph /\ ( ps /\ ph ) ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) )
15	4, 14	mpbi 220	1 |- ( ( ps /\ ph ) -> ch )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037

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  df-3an 1039

This theorem is referenced by: (None)