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Theorem ex-natded9.20 21686
Description: Theorem 9.20 of [Clemente] 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 446 1
311  ( ch  \/  th )  ( ph  ->  ( ch  \/  th ) )  /\ER 1 simprd 450 1
44 ...|  ch  ( ( ph  /\  ch )  ->  ch ) ND hypothesis assumption simpr 448
55 ...  ( ps  /\  ch )  ( ( ph  /\  ch )  ->  ( ps  /\  ch ) )  /\I 2,4 jca 519 3,4
66 ...  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ( ph  /\  ch )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/IR 5 orcd 382 5
78 ...|  th  ( ( ph  /\  th )  ->  th ) ND hypothesis assumption simpr 448
89 ...  ( ps  /\  th )  ( ( ph  /\  th )  ->  ( ps  /\  th ) )  /\I 2,7 jca 519 7,8
910 ...  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ( ph  /\  th )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/IL 8 olcd 383 9
1012  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/E 3,6,9 mpjaodan 762 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 452; simpr 448 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 21687. (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 446 . . . . 5  |-  ( ph  ->  ps )
32adantr 452 . . . 4  |-  ( (
ph  /\  ch )  ->  ps )
4 simpr 448 . . . 4  |-  ( (
ph  /\  ch )  ->  ch )
53, 4jca 519 . . 3  |-  ( (
ph  /\  ch )  ->  ( ps  /\  ch ) )
65orcd 382 . 2  |-  ( (
ph  /\  ch )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
72adantr 452 . . . 4  |-  ( (
ph  /\  th )  ->  ps )
8 simpr 448 . . . 4  |-  ( (
ph  /\  th )  ->  th )
97, 8jca 519 . . 3  |-  ( (
ph  /\  th )  ->  ( ps  /\  th ) )
109olcd 383 . 2  |-  ( (
ph  /\  th )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
111simprd 450 . 2  |-  ( ph  ->  ( ch  \/  th ) )
126, 10, 11mpjaodan 762 1  |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359
This theorem is referenced by:  nobnddown  25577
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 178  df-or 360  df-an 361
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