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Theorem ex-natded9.20 20778
Description: Theorem 9.20 of [Laboreo] p. 43, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
11  ( ps  /\  ( ch  \/  th ) )  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) ) Given $e
22  ps  ( ph  ->  ps )  /\EL 1 simpld 447 1
311  ( ch  \/  th )  ( ph  ->  ( ch  \/  th ) )  /\ER 1 simprd 451 1
44 ...|  ch  ( ( ph  /\  ch )  ->  ch ) ND hypothesis assumption simpr 449
55 ...  ( ps  /\  ch )  ( ( ph  /\  ch )  ->  ( ps  /\  ch ) )  /\I 2,4 jca 520 3,4
66 ...  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ( ph  /\  ch )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/IR 5 orcd 383 5
78 ...|  th  ( ( ph  /\  th )  ->  th ) ND hypothesis assumption simpr 449
89 ...  ( ps  /\  th )  ( ( ph  /\  th )  ->  ( ps  /\  th ) )  /\I 2,7 jca 520 7,8
910 ...  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ( ph  /\  th )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/IL 8 olcd 384 9
1012  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/E 3,6,9 mpjaodan 764 6,10,11

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 453; simpr 449 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof is ex-natded9.20-2 20779. (Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

Hypothesis
Ref Expression
ex-natded9.20.1  |-  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) )
Assertion
Ref Expression
ex-natded9.20  |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )

Proof of Theorem ex-natded9.20
StepHypRef Expression
1 ex-natded9.20.1 . . . . . 6  |-  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) )
21simpld 447 . . . . 5  |-  ( ph  ->  ps )
32adantr 453 . . . 4  |-  ( (
ph  /\  ch )  ->  ps )
4 simpr 449 . . . 4  |-  ( (
ph  /\  ch )  ->  ch )
53, 4jca 520 . . 3  |-  ( (
ph  /\  ch )  ->  ( ps  /\  ch ) )
65orcd 383 . 2  |-  ( (
ph  /\  ch )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
72adantr 453 . . . 4  |-  ( (
ph  /\  th )  ->  ps )
8 simpr 449 . . . 4  |-  ( (
ph  /\  th )  ->  th )
97, 8jca 520 . . 3  |-  ( (
ph  /\  th )  ->  ( ps  /\  th ) )
109olcd 384 . 2  |-  ( (
ph  /\  th )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
111simprd 451 . 2  |-  ( ph  ->  ( ch  \/  th ) )
126, 10, 11mpjaodan 764 1  |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362
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