Theorem uun2221 39040

Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 30-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
uun2221.1	|- ( ( ph /\ ph /\ ( ps /\ ph ) ) -> ch )

Assertion

Ref	Expression
uun2221	|- ( ( ps /\ ph ) -> ch )

Proof of Theorem uun2221

Step	Hyp	Ref	Expression
1		uun2221.1	. 2 |- ( ( ph /\ ph /\ ( ps /\ ph ) ) -> ch )
2		3anass 1042	. . . . . 6 |- ( ( ph /\ ph /\ ( ps /\ ph ) ) <-> ( ph /\ ( ph /\ ( ps /\ ph ) ) ) )
3		anabs5 851	. . . . . 6 |- ( ( ph /\ ( ph /\ ( ps /\ ph ) ) ) <-> ( ph /\ ( ps /\ ph ) ) )
4	2, 3	bitri 264	. . . . 5 |- ( ( ph /\ ph /\ ( ps /\ ph ) ) <-> ( ph /\ ( ps /\ ph ) ) )
5		ancom 466	. . . . . 6 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
6	5	anbi2i 730	. . . . 5 |- ( ( ph /\ ( ph /\ ps ) ) <-> ( ph /\ ( ps /\ ph ) ) )
7	4, 6	bitr4i 267	. . . 4 |- ( ( ph /\ ph /\ ( ps /\ ph ) ) <-> ( ph /\ ( ph /\ ps ) ) )
8		anabs5 851	. . . . 5 |- ( ( ph /\ ( ph /\ ps ) ) <-> ( ph /\ ps ) )
9	8, 5	bitri 264	. . . 4 |- ( ( ph /\ ( ph /\ ps ) ) <-> ( ps /\ ph ) )
10	7, 9	bitri 264	. . 3 |- ( ( ph /\ ph /\ ( ps /\ ph ) ) <-> ( ps /\ ph ) )
11	10	imbi1i 339	. 2 |- ( ( ( ph /\ ph /\ ( ps /\ ph ) ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) )
12	1, 11	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)