Theorem uun2221p1

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.)

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

Proof

Step	Hyp	Ref	Expression
1		uun2221p1.1	|- ( ( ph /\ ( ps /\ ph ) /\ ph ) -> ch )
2		3anrot	|- ( ( ph /\ ph /\ ( ps /\ ph ) ) <-> ( ph /\ ( ps /\ ph ) /\ ph ) )
3	2	imbi1i	|- ( ( ( ph /\ ph /\ ( ps /\ ph ) ) -> ch ) <-> ( ( ph /\ ( ps /\ ph ) /\ ph ) -> ch ) )
4	1 3	mpbir	|- ( ( ph /\ ph /\ ( ps /\ ph ) ) -> ch )
5		3anass	|- ( ( ph /\ ph /\ ( ps /\ ph ) ) <-> ( ph /\ ( ph /\ ( ps /\ ph ) ) ) )
6		anabs5	|- ( ( ph /\ ( ph /\ ( ps /\ ph ) ) ) <-> ( ph /\ ( ps /\ ph ) ) )
7	5 6	bitri	|- ( ( ph /\ ph /\ ( ps /\ ph ) ) <-> ( ph /\ ( ps /\ ph ) ) )
8		ancom	|- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
9	8	anbi2i	|- ( ( ph /\ ( ph /\ ps ) ) <-> ( ph /\ ( ps /\ ph ) ) )
10	7 9	bitr4i	|- ( ( ph /\ ph /\ ( ps /\ ph ) ) <-> ( ph /\ ( ph /\ ps ) ) )
11		anabs5	|- ( ( ph /\ ( ph /\ ps ) ) <-> ( ph /\ ps ) )
12	11 8	bitri	|- ( ( ph /\ ( ph /\ ps ) ) <-> ( ps /\ ph ) )
13	10 12	bitri	|- ( ( ph /\ ph /\ ( ps /\ ph ) ) <-> ( ps /\ ph ) )
14	13	imbi1i	|- ( ( ( ph /\ ph /\ ( ps /\ ph ) ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) )
15	4 14	mpbi	|- ( ( ps /\ ph ) -> ch )