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Theorem ex-natded5.13 20802
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 20803. 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 451 to move it into the ND hypothesis
39  ( -.  ta  ->  -.  ch )  ( ph  ->  ( -.  ta  ->  -.  ch ) ) Given $e. ad2antrr 706 to move it into the ND sub-hypothesis
41 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption simpr 447
54 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 14 1,3
65 ...  ( th  \/  ta )  ( ( ph  /\  ps )  ->  ( th  \/  ta ) )  \/I 5 orcd 381 4
76 ...|  ch  ( ( ph  /\  ch )  ->  ch ) ND hypothesis assumption simpr 447
88 ... ...|  -.  ta  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ta ) (sub) ND hypothesis assumption simpr 447
911 ... ...  -.  ch  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ch )  ->E 3,8 mpd 14 8,10
107 ... ...  ch  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  ch ) IT 7 adantr 451 6
1112 ...  -.  -.  ta  ( ( ph  /\  ch )  ->  -.  -.  ta )  -.I 8,9,10 pm2.65da 559 7,11
1213 ...  ta  ( ( ph  /\  ch )  ->  ta )  -.E 11 notnotrd 105 12
1314 ...  ( th  \/  ta )  ( ( ph  /\  ch )  ->  ( th  \/  ta ) )  \/I 12 olcd 382 13
1416  ( th  \/  ta )  ( ph  ->  ( th  \/  ta ) )  \/E 1,6,13 mpjaodan 761 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 451; simpr 447 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 447 . . . 4  |-  ( (
ph  /\  ps )  ->  ps )
2 ex-natded5.13.2 . . . . 5  |-  ( ph  ->  ( ps  ->  th )
)
32adantr 451 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ps  ->  th )
)
41, 3mpd 14 . . 3  |-  ( (
ph  /\  ps )  ->  th )
54orcd 381 . 2  |-  ( (
ph  /\  ps )  ->  ( th  \/  ta ) )
6 simpr 447 . . . . . 6  |-  ( (
ph  /\  ch )  ->  ch )
76adantr 451 . . . . 5  |-  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  ch )
8 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ta )
9 ex-natded5.13.3 . . . . . . 7  |-  ( ph  ->  ( -.  ta  ->  -. 
ch ) )
109ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  ( -.  ta  ->  -. 
ch ) )
118, 10mpd 14 . . . . 5  |-  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ch )
127, 11pm2.65da 559 . . . 4  |-  ( (
ph  /\  ch )  ->  -.  -.  ta )
1312notnotrd 105 . . 3  |-  ( (
ph  /\  ch )  ->  ta )
1413olcd 382 . 2  |-  ( (
ph  /\  ch )  ->  ( th  \/  ta ) )
15 ex-natded5.13.1 . 2  |-  ( ph  ->  ( ps  \/  ch ) )
165, 14, 15mpjaodan 761 1  |-  ( ph  ->  ( th  \/  ta ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358
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 177  df-or 359  df-an 360
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