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Theorem jaoi2 28176
Description: If one part of a disjunction is already true, the other part can be true only if the first part is false. (Contributed by Alexander van der Vekens, 3-Nov-2017.)
Hypothesis
Ref Expression
jaoi2.1  |-  ( (
ph  \/  ( -.  ph 
/\  ch ) )  ->  ps )
Assertion
Ref Expression
jaoi2  |-  ( (
ph  \/  ch )  ->  ps )

Proof of Theorem jaoi2
StepHypRef Expression
1 exmid 404 . . . 4  |-  ( ph  \/  -.  ph )
2 iba 489 . . . . 5  |-  ( (
ph  \/  -.  ph )  ->  ( ch  <->  ( ch  /\  ( ph  \/  -.  ph ) ) ) )
3 ancom 437 . . . . . 6  |-  ( ( ch  /\  ( ph  \/  -.  ph ) )  <-> 
( ( ph  \/  -.  ph )  /\  ch ) )
4 andir 838 . . . . . 6  |-  ( ( ( ph  \/  -.  ph )  /\  ch )  <->  ( ( ph  /\  ch )  \/  ( -.  ph 
/\  ch ) ) )
53, 4bitri 240 . . . . 5  |-  ( ( ch  /\  ( ph  \/  -.  ph ) )  <-> 
( ( ph  /\  ch )  \/  ( -.  ph  /\  ch )
) )
62, 5syl6bb 252 . . . 4  |-  ( (
ph  \/  -.  ph )  ->  ( ch  <->  ( ( ph  /\  ch )  \/  ( -.  ph  /\  ch ) ) ) )
71, 6ax-mp 8 . . 3  |-  ( ch  <->  ( ( ph  /\  ch )  \/  ( -.  ph 
/\  ch ) ) )
87orbi2i 505 . 2  |-  ( (
ph  \/  ch )  <->  (
ph  \/  ( ( ph  /\  ch )  \/  ( -.  ph  /\  ch ) ) ) )
9 orass 510 . . . . 5  |-  ( ( ( ph  \/  ( ph  /\  ch ) )  \/  ( -.  ph  /\ 
ch ) )  <->  ( ph  \/  ( ( ph  /\  ch )  \/  ( -.  ph  /\  ch )
) ) )
109bicomi 193 . . . 4  |-  ( (
ph  \/  ( ( ph  /\  ch )  \/  ( -.  ph  /\  ch ) ) )  <->  ( ( ph  \/  ( ph  /\  ch ) )  \/  ( -.  ph  /\  ch )
) )
11 pm4.44 560 . . . . . 6  |-  ( ph  <->  (
ph  \/  ( ph  /\ 
ch ) ) )
1211bicomi 193 . . . . 5  |-  ( (
ph  \/  ( ph  /\ 
ch ) )  <->  ph )
1312orbi1i 506 . . . 4  |-  ( ( ( ph  \/  ( ph  /\  ch ) )  \/  ( -.  ph  /\ 
ch ) )  <->  ( ph  \/  ( -.  ph  /\  ch ) ) )
1410, 13bitri 240 . . 3  |-  ( (
ph  \/  ( ( ph  /\  ch )  \/  ( -.  ph  /\  ch ) ) )  <->  ( ph  \/  ( -.  ph  /\  ch ) ) )
15 jaoi2.1 . . 3  |-  ( (
ph  \/  ( -.  ph 
/\  ch ) )  ->  ps )
1614, 15sylbi 187 . 2  |-  ( (
ph  \/  ( ( ph  /\  ch )  \/  ( -.  ph  /\  ch ) ) )  ->  ps )
178, 16sylbi 187 1  |-  ( (
ph  \/  ch )  ->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358
This theorem is referenced by:  trlonprop  28341
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-or 359  df-an 360
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