Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pm2.43cbi Unicode version

Theorem pm2.43cbi 28579
Description: Logical equivalence of a 3-left-nested implication and a 2-left-nested implicated when two antecedents of the former implication are identical. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
1::  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  )  ->  ( ph  ->  ( ps  ->  ( ph  ->  ( ch  ->  th ) ) ) ) )
2::  |-  ( ( ph  ->  ( ps  ->  ( ph  ->  ( ch  ->  th ) ) )  )  ->  ( ps  ->  ( ph  ->  ( ch  ->  th ) ) ) )
3:1,2:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  )  ->  ( ps  ->  ( ph  ->  ( ch  ->  th ) ) ) )
4::  |-  ( ( ps  ->  ( ph  ->  ( ch  ->  th ) ) )  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
5:3,4:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  )  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
6::  |-  ( ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) ) )
qed:5,6:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )  )  <->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
Assertion
Ref Expression
pm2.43cbi  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )  <->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )

Proof of Theorem pm2.43cbi
StepHypRef Expression
1 pm2.24 101 . . . 4  |-  ( ph  ->  ( -.  ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
21com4l 78 . . 3  |-  ( -. 
ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
3 id 19 . . 3  |-  ( ( ps  ->  ( ch  ->  ( ph  ->  th )
) )  ->  ( ps  ->  ( ch  ->  (
ph  ->  th ) ) ) )
42, 3ja 153 . 2  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
5 ax-1 5 . 2  |-  ( ( ps  ->  ( ch  ->  ( ph  ->  th )
) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) ) )
64, 5impbii 180 1  |-  ( (
ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )  <->  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176
This theorem is referenced by:  ee233  28580  ee33VD  28971
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
  Copyright terms: Public domain W3C validator