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Theorem ex-natded5.7 20798
Description: Theorem 5.7 of [Clemente] p. 19, translated line by line using the interpretation of natural deduction in Metamath. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.7-2 20799. 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:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
16  ( ps  \/  ( ch  /\  th ) )  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) ) Given $e. No need for adantr 451 because we do not move this into an ND hypothesis
21 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption (new scope) simpr 447
32 ...  ( ps  \/  ch )  ( ( ph  /\  ps )  ->  ( ps  \/  ch ) )  \/IL 2 orcd 381, the MPE equivalent of  \/IL, 1
43 ...|  ( ch  /\  th )  ( ( ph  /\  ( ch  /\  th ) )  ->  ( ch  /\  th ) ) ND hypothesis assumption (new scope) simpr 447
54 ...  ch  ( ( ph  /\  ( ch  /\  th ) )  ->  ch )  /\EL 4 simpld 445, the MPE equivalent of  /\EL, 3
66 ...  ( ps  \/  ch )  ( ( ph  /\  ( ch  /\  th ) )  ->  ( ps  \/  ch ) )  \/IR 5 olcd 382, the MPE equivalent of  \/IR, 4
77  ( ps  \/  ch )  ( ph  ->  ( ps  \/  ch ) )  \/E 1,3,6 mpjaodan 761, the MPE equivalent of  \/E, 2,5,6

The original used Latin letters for predicates; 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. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Hypothesis
Ref Expression
ex-natded5.7.1  |-  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) )
Assertion
Ref Expression
ex-natded5.7  |-  ( ph  ->  ( ps  \/  ch ) )

Proof of Theorem ex-natded5.7
StepHypRef Expression
1 simpr 447 . . 3  |-  ( (
ph  /\  ps )  ->  ps )
21orcd 381 . 2  |-  ( (
ph  /\  ps )  ->  ( ps  \/  ch ) )
3 simpr 447 . . . 4  |-  ( (
ph  /\  ( ch  /\ 
th ) )  -> 
( ch  /\  th ) )
43simpld 445 . . 3  |-  ( (
ph  /\  ( ch  /\ 
th ) )  ->  ch )
54olcd 382 . 2  |-  ( (
ph  /\  ( ch  /\ 
th ) )  -> 
( ps  \/  ch ) )
6 ex-natded5.7.1 . 2  |-  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) )
72, 5, 6mpjaodan 761 1  |-  ( ph  ->  ( ps  \/  ch ) )
Colors of variables: wff set class
Syntax hints:    -> 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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