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Theorem imbi12 323
Description: Closed form of imbi12i 327. Was automatically derived from its "Virtual Deduction" version and Metamath's "minimize" command. (Contributed by Alan Sare, 18-Mar-2012.)
Assertion
Ref Expression
imbi12  |-  ( (
ph 
<->  ps )  ->  (
( ch  <->  th )  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ) )

Proof of Theorem imbi12
StepHypRef Expression
1 simplim 154 . . 3  |-  ( -.  ( ( ph  <->  ps )  ->  -.  ( ch  <->  th )
)  ->  ( ph  <->  ps ) )
2 simprim 153 . . 3  |-  ( -.  ( ( ph  <->  ps )  ->  -.  ( ch  <->  th )
)  ->  ( ch  <->  th ) )
31, 2imbi12d 321 . 2  |-  ( -.  ( ( ph  <->  ps )  ->  -.  ( ch  <->  th )
)  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) )
43expi 152 1  |-  ( (
ph 
<->  ps )  ->  (
( ch  <->  th )  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187
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 188
This theorem is referenced by:  imbi12i  327  bj-imbi12  30949  ifpbi12  35831  ifpbi13  35832  imbi13  36514  imbi13VD  36911  sbcssgVD  36920
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