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Metamath Proof Explorer

Theorem un2122

Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis un2122.1 
|- ( ( ( ph /\ ps ) /\ ps /\ ps ) -> ch )
Assertion un2122 
|- ( ( ph /\ ps ) -> ch )

Proof

Step Hyp Ref Expression
1 un2122.1 
 |-  ( ( ( ph /\ ps ) /\ ps /\ ps ) -> ch )
2 3anass 
 |-  ( ( ( ph /\ ps ) /\ ps /\ ps ) <-> ( ( ph /\ ps ) /\ ( ps /\ ps ) ) )
3 anandir 
 |-  ( ( ( ph /\ ps ) /\ ps ) <-> ( ( ph /\ ps ) /\ ( ps /\ ps ) ) )
4 ancom 
 |-  ( ( ( ph /\ ps ) /\ ps ) <-> ( ps /\ ( ph /\ ps ) ) )
5 anabs7 
 |-  ( ( ps /\ ( ph /\ ps ) ) <-> ( ph /\ ps ) )
6 4 5 bitri 
 |-  ( ( ( ph /\ ps ) /\ ps ) <-> ( ph /\ ps ) )
7 3 6 bitr3i 
 |-  ( ( ( ph /\ ps ) /\ ( ps /\ ps ) ) <-> ( ph /\ ps ) )
8 2 7 bitri 
 |-  ( ( ( ph /\ ps ) /\ ps /\ ps ) <-> ( ph /\ ps ) )
9 8 1 sylbir 
 |-  ( ( ph /\ ps ) -> ch )