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Theorem ex-natded5.13 20824
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 20825. 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 453 to move it into the ND hypothesis
39  ( -.  ta  ->  -.  ch )  ( ph  ->  ( -.  ta  ->  -.  ch ) ) Given $e. ad2antrr 708 to move it into the ND sub-hypothesis
41 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption simpr 449
54 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 16 1,3
65 ...  ( th  \/  ta )  ( ( ph  /\  ps )  ->  ( th  \/  ta ) )  \/I 5 orcd 383 4
76 ...|  ch  ( ( ph  /\  ch )  ->  ch ) ND hypothesis assumption simpr 449
88 ... ...|  -.  ta  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ta ) (sub) ND hypothesis assumption simpr 449
911 ... ...  -.  ch  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ch )  ->E 3,8 mpd 16 8,10
107 ... ...  ch  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  ch ) IT 7 adantr 453 6
1112 ...  -.  -.  ta  ( ( ph  /\  ch )  ->  -.  -.  ta )  -.I 8,9,10 pm2.65da 561 7,11
1213 ...  ta  ( ( ph  /\  ch )  ->  ta )  -.E 11 notnotrd 107 12
1314 ...  ( th  \/  ta )  ( ( ph  /\  ch )  ->  ( th  \/  ta ) )  \/I 12 olcd 384 13
1416  ( th  \/  ta )  ( ph  ->  ( th  \/  ta ) )  \/E 1,6,13 mpjaodan 763 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 453; simpr 449 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 449 . . . 4  |-  ( (
ph  /\  ps )  ->  ps )
2 ex-natded5.13.2 . . . . 5  |-  ( ph  ->  ( ps  ->  th )
)
32adantr 453 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ps  ->  th )
)
41, 3mpd 16 . . 3  |-  ( (
ph  /\  ps )  ->  th )
54orcd 383 . 2  |-  ( (
ph  /\  ps )  ->  ( th  \/  ta ) )
6 simpr 449 . . . . . 6  |-  ( (
ph  /\  ch )  ->  ch )
76adantr 453 . . . . 5  |-  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  ch )
8 simpr 449 . . . . . 6  |-  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ta )
9 ex-natded5.13.3 . . . . . . 7  |-  ( ph  ->  ( -.  ta  ->  -. 
ch ) )
109ad2antrr 708 . . . . . 6  |-  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  ( -.  ta  ->  -. 
ch ) )
118, 10mpd 16 . . . . 5  |-  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ch )
127, 11pm2.65da 561 . . . 4  |-  ( (
ph  /\  ch )  ->  -.  -.  ta )
1312notnotrd 107 . . 3  |-  ( (
ph  /\  ch )  ->  ta )
1413olcd 384 . 2  |-  ( (
ph  /\  ch )  ->  ( th  \/  ta ) )
15 ex-natded5.13.1 . 2  |-  ( ph  ->  ( ps  \/  ch ) )
165, 14, 15mpjaodan 763 1  |-  ( ph  ->  ( th  \/  ta ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> 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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