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Theorem un2122 28565
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.)
Hypothesis
Ref Expression
un2122.1  |-  ( ( ( ph  /\  ps )  /\  ps  /\  ps )  ->  ch )
Assertion
Ref Expression
un2122  |-  ( (
ph  /\  ps )  ->  ch )

Proof of Theorem un2122
StepHypRef Expression
1 3anass 938 . . 3  |-  ( ( ( ph  /\  ps )  /\  ps  /\  ps ) 
<->  ( ( ph  /\  ps )  /\  ( ps  /\  ps ) ) )
2 anandir 802 . . . 4  |-  ( ( ( ph  /\  ps )  /\  ps )  <->  ( ( ph  /\  ps )  /\  ( ps  /\  ps )
) )
3 ancom 437 . . . . 5  |-  ( ( ( ph  /\  ps )  /\  ps )  <->  ( ps  /\  ( ph  /\  ps ) ) )
4 anabs7 785 . . . . 5  |-  ( ( ps  /\  ( ph  /\ 
ps ) )  <->  ( ph  /\ 
ps ) )
53, 4bitri 240 . . . 4  |-  ( ( ( ph  /\  ps )  /\  ps )  <->  ( ph  /\ 
ps ) )
62, 5bitr3i 242 . . 3  |-  ( ( ( ph  /\  ps )  /\  ( ps  /\  ps ) )  <->  ( ph  /\ 
ps ) )
71, 6bitri 240 . 2  |-  ( ( ( ph  /\  ps )  /\  ps  /\  ps ) 
<->  ( ph  /\  ps ) )
8 un2122.1 . 2  |-  ( ( ( ph  /\  ps )  /\  ps  /\  ps )  ->  ch )
97, 8sylbir 204 1  |-  ( (
ph  /\  ps )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934
This theorem is referenced by:  suctrALT3  28700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936
  Copyright terms: Public domain W3C validator