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Theorem ex-natded5.13 21706
Description: Theorem 5.13 of [Clemente] p. 20, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.13-2 21707. 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
115  ( ps  \/  ch )  ( ph  ->  ( ps  \/  ch ) ) Given $e.
2;32  ( ps  ->  th )  ( ph  ->  ( ps  ->  th ) ) Given $e. adantr 452 to move it into the ND hypothesis
39  ( -.  ta  ->  -.  ch )  ( ph  ->  ( -.  ta  ->  -.  ch ) ) Given $e. ad2antrr 707 to move it into the ND sub-hypothesis
41 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption simpr 448
54 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 15 1,3
65 ...  ( th  \/  ta )  ( ( ph  /\  ps )  ->  ( th  \/  ta ) )  \/I 5 orcd 382 4
76 ...|  ch  ( ( ph  /\  ch )  ->  ch ) ND hypothesis assumption simpr 448
88 ... ...|  -.  ta  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ta ) (sub) ND hypothesis assumption simpr 448
911 ... ...  -.  ch  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ch )  ->E 3,8 mpd 15 8,10
107 ... ...  ch  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  ch ) IT 7 adantr 452 6
1112 ...  -.  -.  ta  ( ( ph  /\  ch )  ->  -.  -.  ta )  -.I 8,9,10 pm2.65da 560 7,11
1213 ...  ta  ( ( ph  /\  ch )  ->  ta )  -.E 11 notnotrd 107 12
1314 ...  ( th  \/  ta )  ( ( ph  /\  ch )  ->  ( th  \/  ta ) )  \/I 12 olcd 383 13
1416  ( th  \/  ta )  ( ph  ->  ( th  \/  ta ) )  \/E 1,6,13 mpjaodan 762 5,14,15

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). (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Hypotheses
Ref Expression
ex-natded5.13.1  |-  ( ph  ->  ( ps  \/  ch ) )
ex-natded5.13.2  |-  ( ph  ->  ( ps  ->  th )
)
ex-natded5.13.3  |-  ( ph  ->  ( -.  ta  ->  -. 
ch ) )
Assertion
Ref Expression
ex-natded5.13  |-  ( ph  ->  ( th  \/  ta ) )

Proof of Theorem ex-natded5.13
StepHypRef Expression
1 simpr 448 . . . 4  |-  ( (
ph  /\  ps )  ->  ps )
2 ex-natded5.13.2 . . . . 5  |-  ( ph  ->  ( ps  ->  th )
)
32adantr 452 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ps  ->  th )
)
41, 3mpd 15 . . 3  |-  ( (
ph  /\  ps )  ->  th )
54orcd 382 . 2  |-  ( (
ph  /\  ps )  ->  ( th  \/  ta ) )
6 simpr 448 . . . . . 6  |-  ( (
ph  /\  ch )  ->  ch )
76adantr 452 . . . . 5  |-  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  ch )
8 simpr 448 . . . . . 6  |-  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ta )
9 ex-natded5.13.3 . . . . . . 7  |-  ( ph  ->  ( -.  ta  ->  -. 
ch ) )
109ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  ( -.  ta  ->  -. 
ch ) )
118, 10mpd 15 . . . . 5  |-  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ch )
127, 11pm2.65da 560 . . . 4  |-  ( (
ph  /\  ch )  ->  -.  -.  ta )
1312notnotrd 107 . . 3  |-  ( (
ph  /\  ch )  ->  ta )
1413olcd 383 . 2  |-  ( (
ph  /\  ch )  ->  ( th  \/  ta ) )
15 ex-natded5.13.1 . 2  |-  ( ph  ->  ( ps  \/  ch ) )
165, 14, 15mpjaodan 762 1  |-  ( ph  ->  ( th  \/  ta ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359
This theorem is referenced by:  ex-natded5.13-2  21707
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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