Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ad5ant23 Unicode version

Theorem ad5ant23 28531
Description: Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
Hypothesis
Ref Expression
ad5ant23.1  |-  ( (
ph  /\  ps )  ->  ch )
Assertion
Ref Expression
ad5ant23  |-  ( ( ( ( ( th 
/\  ph )  /\  ps )  /\  ta )  /\  et )  ->  ch )

Proof of Theorem ad5ant23
StepHypRef Expression
1 ad5ant23.1 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  ch )
21ex 423 . . . . . . . 8  |-  ( ph  ->  ( ps  ->  ch ) )
32a1i34 26307 . . . . . . 7  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ta  ->  ch ) ) ) )
43a1i4 26304 . . . . . 6  |-  ( ph  ->  ( ps  ->  ( th  ->  ( et  ->  ( ta  ->  ch )
) ) ) )
54com45 83 . . . . 5  |-  ( ph  ->  ( ps  ->  ( th  ->  ( ta  ->  ( et  ->  ch )
) ) ) )
65com23 72 . . . 4  |-  ( ph  ->  ( th  ->  ( ps  ->  ( ta  ->  ( et  ->  ch )
) ) ) )
76com12 27 . . 3  |-  ( th 
->  ( ph  ->  ( ps  ->  ( ta  ->  ( et  ->  ch )
) ) ) )
87imp 418 . 2  |-  ( ( th  /\  ph )  ->  ( ps  ->  ( ta  ->  ( et  ->  ch ) ) ) )
98imp41 576 1  |-  ( ( ( ( ( th 
/\  ph )  /\  ps )  /\  ta )  /\  et )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358
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
  Copyright terms: Public domain W3C validator