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Theorem cnf1dd 29061
Description: A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
Hypotheses
Ref Expression
cnf1dd.1  |-  ( ph  ->  ( ps  ->  -.  ch ) )
cnf1dd.2  |-  ( ph  ->  ( ps  ->  ( ch  \/  th ) ) )
Assertion
Ref Expression
cnf1dd  |-  ( ph  ->  ( ps  ->  th )
)

Proof of Theorem cnf1dd
StepHypRef Expression
1 cnf1dd.1 . . 3  |-  ( ph  ->  ( ps  ->  -.  ch ) )
2 cnf1dd.2 . . 3  |-  ( ph  ->  ( ps  ->  ( ch  \/  th ) ) )
31, 2jcad 533 . 2  |-  ( ph  ->  ( ps  ->  ( -.  ch  /\  ( ch  \/  th ) ) ) )
4 df-or 370 . . . . . 6  |-  ( ( ch  \/  th )  <->  ( -.  ch  ->  th )
)
54anbi2i 694 . . . . 5  |-  ( ( -.  ch  /\  ( ch  \/  th ) )  <-> 
( -.  ch  /\  ( -.  ch  ->  th ) ) )
6 pm3.35 587 . . . . 5  |-  ( ( -.  ch  /\  ( -.  ch  ->  th )
)  ->  th )
75, 6sylbi 195 . . . 4  |-  ( ( -.  ch  /\  ( ch  \/  th ) )  ->  th )
87imim2i 14 . . 3  |-  ( ( ps  ->  ( -.  ch  /\  ( ch  \/  th ) ) )  -> 
( ps  ->  th )
)
98imim2i 14 . 2  |-  ( (
ph  ->  ( ps  ->  ( -.  ch  /\  ( ch  \/  th ) ) ) )  ->  ( ph  ->  ( ps  ->  th ) ) )
103, 9ax-mp 5 1  |-  ( ph  ->  ( ps  ->  th )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369
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 370  df-an 371
This theorem is referenced by:  cnf2dd  29062  cnfn1dd  29063  mpt2bi123f  29143  mptbi12f  29147  ac6s6  29152
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