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Theorem dfvd3 28610
Description: Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dfvd3  |-  ( (.
ph ,. ps ,. ch  ->.  th ).  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )

Proof of Theorem dfvd3
StepHypRef Expression
1 df-vd3 28609 . 2  |-  ( (.
ph ,. ps ,. ch  ->.  th ).  <->  ( ( ph  /\  ps  /\  ch )  ->  th ) )
2 df-3an 938 . . . . 5  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
32imbi1i 316 . . . 4  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  th )  <->  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) )
4 impexp 434 . . . 4  |-  ( ( ( ( ph  /\  ps )  /\  ch )  ->  th )  <->  ( ( ph  /\  ps )  -> 
( ch  ->  th )
) )
53, 4bitri 241 . . 3  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  th )  <->  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) )
6 impexp 434 . . 3  |-  ( ( ( ph  /\  ps )  ->  ( ch  ->  th ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
75, 6bitri 241 . 2  |-  ( ( ( ph  /\  ps  /\ 
ch )  ->  th )  <->  (
ph  ->  ( ps  ->  ( ch  ->  th )
) ) )
81, 7bitri 241 1  |-  ( (.
ph ,. ps ,. ch  ->.  th ).  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   (.wvd3 28606
This theorem is referenced by:  dfvd3i  28611  dfvd3ir  28612
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 178  df-an 361  df-3an 938  df-vd3 28609
  Copyright terms: Public domain W3C validator