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Theorem uun2221p1 31902
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
StepHypRef Expression
1 uun2221p1.1 . . 3  |-  ( (
ph  /\  ( ps  /\ 
ph )  /\  ph )  ->  ch )
2 3anrot 970 . . . 4  |-  ( (
ph  /\  ph  /\  ( ps  /\  ph ) )  <-> 
( ph  /\  ( ps  /\  ph )  /\  ph ) )
32imbi1i 325 . . 3  |-  ( ( ( ph  /\  ph  /\  ( ps  /\  ph ) )  ->  ch ) 
<->  ( ( ph  /\  ( ps  /\  ph )  /\  ph )  ->  ch ) )
41, 3mpbir 209 . 2  |-  ( (
ph  /\  ph  /\  ( ps  /\  ph ) )  ->  ch )
5 3anass 969 . . . . . 6  |-  ( (
ph  /\  ph  /\  ( ps  /\  ph ) )  <-> 
( ph  /\  ( ph  /\  ( ps  /\  ph ) ) ) )
6 anabs5 807 . . . . . 6  |-  ( (
ph  /\  ( ph  /\  ( ps  /\  ph ) ) )  <->  ( ph  /\  ( ps  /\  ph ) ) )
75, 6bitri 249 . . . . 5  |-  ( (
ph  /\  ph  /\  ( ps  /\  ph ) )  <-> 
( ph  /\  ( ps  /\  ph ) ) )
8 ancom 450 . . . . . 6  |-  ( (
ph  /\  ps )  <->  ( ps  /\  ph )
)
98anbi2i 694 . . . . 5  |-  ( (
ph  /\  ( ph  /\ 
ps ) )  <->  ( ph  /\  ( ps  /\  ph ) ) )
107, 9bitr4i 252 . . . 4  |-  ( (
ph  /\  ph  /\  ( ps  /\  ph ) )  <-> 
( ph  /\  ( ph  /\  ps ) ) )
11 anabs5 807 . . . . 5  |-  ( (
ph  /\  ( ph  /\ 
ps ) )  <->  ( ph  /\ 
ps ) )
1211, 8bitri 249 . . . 4  |-  ( (
ph  /\  ( ph  /\ 
ps ) )  <->  ( ps  /\ 
ph ) )
1310, 12bitri 249 . . 3  |-  ( (
ph  /\  ph  /\  ( ps  /\  ph ) )  <-> 
( ps  /\  ph ) )
1413imbi1i 325 . 2  |-  ( ( ( ph  /\  ph  /\  ( ps  /\  ph ) )  ->  ch ) 
<->  ( ( ps  /\  ph )  ->  ch )
)
154, 14mpbi 208 1  |-  ( ( ps  /\  ph )  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965
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 185  df-an 371  df-3an 967
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator