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Theorem bitr3VD 28625
Description: Virtual deduction proof of bitr3 28272. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ph  <->  ps )  ->.  ( ph  <->  ps ) ).
2:1,?: e1_ 28399  |-  (. ( ph  <->  ps )  ->.  ( ps  <->  ph ) ).
3::  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ph  <->  ch ) ).
4:3,?: e2 28403  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ch  <->  ph ) ).
5:2,4,?: e12 28499  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ps  <->  ch ) ).
6:5:  |-  (. ( ph  <->  ps )  ->.  ( ( ph  <->  ch )  ->  ( ps  <->  ch ) ) ).
qed:6:  |-  ( ( ph  <->  ps )  ->  ( ( ph  <->  ch )  ->  ( ps  <->  ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bitr3VD  |-  ( (
ph 
<->  ps )  ->  (
( ph  <->  ch )  ->  ( ps 
<->  ch ) ) )

Proof of Theorem bitr3VD
StepHypRef Expression
1 id 19 . . 3  |-  ( (
ph 
<->  ps )  ->  ( ph 
<->  ps ) )
21bicomd 192 . 2  |-  ( (
ph 
<->  ps )  ->  ( ps 
<-> 
ph ) )
3 id 19 . . 3  |-  ( (
ph 
<->  ch )  ->  ( ph 
<->  ch ) )
43bicomd 192 . 2  |-  ( (
ph 
<->  ch )  ->  ( ch 
<-> 
ph ) )
5 biantr 897 . . 3  |-  ( ( ( ps  <->  ph )  /\  ( ch  <->  ph ) )  -> 
( ps  <->  ch )
)
65ex 423 . 2  |-  ( ( ps  <->  ph )  ->  (
( ch  <->  ph )  -> 
( ps  <->  ch )
) )
72, 4, 6syl2im 34 1  |-  ( (
ph 
<->  ps )  ->  (
( ph  <->  ch )  ->  ( ps 
<->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176
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