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Theorem ex-natded5.7 21707
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 21708. 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 452 because we do not move this into an ND hypothesis
21 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption (new scope) simpr 448
32 ...  ( ps  \/  ch )  ( ( ph  /\  ps )  ->  ( ps  \/  ch ) )  \/IL 2 orcd 382, the MPE equivalent of  \/IL, 1
43 ...|  ( ch  /\  th )  ( ( ph  /\  ( ch  /\  th ) )  ->  ( ch  /\  th ) ) ND hypothesis assumption (new scope) simpr 448
54 ...  ch  ( ( ph  /\  ( ch  /\  th ) )  ->  ch )  /\EL 4 simpld 446, the MPE equivalent of  /\EL, 3
66 ...  ( ps  \/  ch )  ( ( ph  /\  ( ch  /\  th ) )  ->  ( ps  \/  ch ) )  \/IR 5 olcd 383, the MPE equivalent of  \/IR, 4
77  ( ps  \/  ch )  ( ph  ->  ( ps  \/  ch ) )  \/E 1,3,6 mpjaodan 762, 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 448 . . 3  |-  ( (
ph  /\  ps )  ->  ps )
21orcd 382 . 2  |-  ( (
ph  /\  ps )  ->  ( ps  \/  ch ) )
3 simpr 448 . . . 4  |-  ( (
ph  /\  ( ch  /\ 
th ) )  -> 
( ch  /\  th ) )
43simpld 446 . . 3  |-  ( (
ph  /\  ( ch  /\ 
th ) )  ->  ch )
54olcd 383 . 2  |-  ( (
ph  /\  ( ch  /\ 
th ) )  -> 
( ps  \/  ch ) )
6 ex-natded5.7.1 . 2  |-  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) )
72, 5, 6mpjaodan 762 1  |-  ( ph  ->  ( ps  \/  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359
This theorem is referenced by:  ex-natded5.7-2  21708
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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