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Theorem cases 979
Description: Case disjunction according to the value of  ph. (Contributed by NM, 25-Apr-2019.)
Hypotheses
Ref Expression
cases.1  |-  ( ph  ->  ( ps  <->  ch )
)
cases.2  |-  ( -. 
ph  ->  ( ps  <->  th )
)
Assertion
Ref Expression
cases  |-  ( ps  <->  ( ( ph  /\  ch )  \/  ( -.  ph 
/\  th ) ) )

Proof of Theorem cases
StepHypRef Expression
1 exmid 416 . . 3  |-  ( ph  \/  -.  ph )
21biantrur 508 . 2  |-  ( ps  <->  ( ( ph  \/  -.  ph )  /\  ps )
)
3 andir 876 . 2  |-  ( ( ( ph  \/  -.  ph )  /\  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ps ) ) )
4 cases.1 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
54pm5.32i 641 . . 3  |-  ( (
ph  /\  ps )  <->  (
ph  /\  ch )
)
6 cases.2 . . . 4  |-  ( -. 
ph  ->  ( ps  <->  th )
)
76pm5.32i 641 . . 3  |-  ( ( -.  ph  /\  ps )  <->  ( -.  ph  /\  th )
)
85, 7orbi12i 523 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ps ) )  <->  ( ( ph  /\  ch )  \/  ( -.  ph  /\  th ) ) )
92, 3, 83bitri 274 1  |-  ( ps  <->  ( ( ph  /\  ch )  \/  ( -.  ph 
/\  th ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370
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  df-or 371  df-an 372
This theorem is referenced by:  casesifp  1434  elimif  3949  elim2if  28000
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