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Theorem bj-con3ALT 32453
Description: Version of con3th 949 using the conditional operator in its proof. (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-con3ALT  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )

Proof of Theorem bj-con3ALT
StepHypRef Expression
1 bicom1 199 . . . 4  |-  ( (if- ( ( ph  ->  ps ) ,  ps ,  ph )  <->  ps )  ->  ( ps 
<-> if- ( ( ph  ->  ps ) ,  ps ,  ph ) ) )
21notbid 294 . . 3  |-  ( (if- ( ( ph  ->  ps ) ,  ps ,  ph )  <->  ps )  ->  ( -.  ps  <->  -. if- ( ( ph  ->  ps ) ,  ps ,  ph )
) )
32imbi1d 317 . 2  |-  ( (if- ( ( ph  ->  ps ) ,  ps ,  ph )  <->  ps )  ->  (
( -.  ps  ->  -. 
ph )  <->  ( -. if- ( ( ph  ->  ps ) ,  ps ,  ph )  ->  -.  ph ) ) )
41imbi2d 316 . . . 4  |-  ( (if- ( ( ph  ->  ps ) ,  ps ,  ph )  <->  ps )  ->  (
( ph  ->  ps )  <->  (
ph  -> if- ( ( ph  ->  ps ) ,  ps ,  ph ) ) ) )
5 bicom1 199 . . . . 5  |-  ( (if- ( ( ph  ->  ps ) ,  ps ,  ph )  <->  ph )  ->  ( ph 
<-> if- ( ( ph  ->  ps ) ,  ps ,  ph ) ) )
65imbi2d 316 . . . 4  |-  ( (if- ( ( ph  ->  ps ) ,  ps ,  ph )  <->  ph )  ->  (
( ph  ->  ph )  <->  (
ph  -> if- ( ( ph  ->  ps ) ,  ps ,  ph ) ) ) )
7 id 22 . . . 4  |-  ( ph  ->  ph )
84, 6, 7bj-elimhyp 32451 . . 3  |-  ( ph  -> if- ( ( ph  ->  ps ) ,  ps ,  ph ) )
98con3i 135 . 2  |-  ( -. if- ( ( ph  ->  ps ) ,  ps ,  ph )  ->  -.  ph )
103, 9bj-dedthm 32452 1  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184  if-wif 32435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371  df-bj-if 32436
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator