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Theorem wl-cases2-dnf 31337
Description: A particular instance of orddi 870 and anddi 871 converting between disjunctive and conjunctive normal forms, when both  ph and  -. 
ph appear. This theorem in fact rephrases cases2 972, and is related to consensus 960. I restate it here in DNF and CNF. The proof deliberately does not use df-ifp 1387 and dfifp4 1390, by which it can be shortened. (Contributed by Wolf Lammen, 21-Jun-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
wl-cases2-dnf  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ch ) )  <->  ( ( -.  ph  \/  ps )  /\  ( ph  \/  ch ) ) )

Proof of Theorem wl-cases2-dnf
StepHypRef Expression
1 exmid 413 . . . . 5  |-  ( ph  \/  -.  ph )
21biantrur 504 . . . 4  |-  ( (
ph  \/  ch )  <->  ( ( ph  \/  -.  ph )  /\  ( ph  \/  ch ) ) )
3 orcom 385 . . . . 5  |-  ( ( -.  ph  \/  ps ) 
<->  ( ps  \/  -.  ph ) )
4 orcom 385 . . . . 5  |-  ( ( ch  \/  ps )  <->  ( ps  \/  ch )
)
53, 4anbi12i 695 . . . 4  |-  ( ( ( -.  ph  \/  ps )  /\  ( ch  \/  ps ) )  <-> 
( ( ps  \/  -.  ph )  /\  ( ps  \/  ch ) ) )
62, 5anbi12i 695 . . 3  |-  ( ( ( ph  \/  ch )  /\  ( ( -. 
ph  \/  ps )  /\  ( ch  \/  ps ) ) )  <->  ( (
( ph  \/  -.  ph )  /\  ( ph  \/  ch ) )  /\  ( ( ps  \/  -.  ph )  /\  ( ps  \/  ch ) ) ) )
7 anass 647 . . 3  |-  ( ( ( ( ph  \/  ch )  /\  ( -.  ph  \/  ps )
)  /\  ( ch  \/  ps ) )  <->  ( ( ph  \/  ch )  /\  ( ( -.  ph  \/  ps )  /\  ( ch  \/  ps ) ) ) )
8 orddi 870 . . 3  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ch ) )  <->  ( (
( ph  \/  -.  ph )  /\  ( ph  \/  ch ) )  /\  ( ( ps  \/  -.  ph )  /\  ( ps  \/  ch ) ) ) )
96, 7, 83bitr4ri 278 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ch ) )  <->  ( (
( ph  \/  ch )  /\  ( -.  ph  \/  ps ) )  /\  ( ch  \/  ps ) ) )
10 wl-orel12 31336 . . 3  |-  ( ( ( ph  \/  ch )  /\  ( -.  ph  \/  ps ) )  -> 
( ch  \/  ps ) )
1110pm4.71i 630 . 2  |-  ( ( ( ph  \/  ch )  /\  ( -.  ph  \/  ps ) )  <->  ( (
( ph  \/  ch )  /\  ( -.  ph  \/  ps ) )  /\  ( ch  \/  ps ) ) )
12 ancom 448 . 2  |-  ( ( ( ph  \/  ch )  /\  ( -.  ph  \/  ps ) )  <->  ( ( -.  ph  \/  ps )  /\  ( ph  \/  ch ) ) )
139, 11, 123bitr2i 273 1  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ch ) )  <->  ( ( -.  ph  \/  ps )  /\  ( ph  \/  ch ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 366    /\ wa 367
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-or 368  df-an 369
This theorem is referenced by: (None)
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