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Theorem ex-natded5.7 20818
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 20819. 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 453 because we do not move this into an ND hypothesis
21 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption (new scope) simpr 449
32 ...  ( ps  \/  ch )  ( ( ph  /\  ps )  ->  ( ps  \/  ch ) )  \/IL 2 orcd 383, the MPE equivalent of  \/IL, 1
43 ...|  ( ch  /\  th )  ( ( ph  /\  ( ch  /\  th ) )  ->  ( ch  /\  th ) ) ND hypothesis assumption (new scope) simpr 449
54 ...  ch  ( ( ph  /\  ( ch  /\  th ) )  ->  ch )  /\EL 4 simpld 447, the MPE equivalent of  /\EL, 3
66 ...  ( ps  \/  ch )  ( ( ph  /\  ( ch  /\  th ) )  ->  ( ps  \/  ch ) )  \/IR 5 olcd 384, the MPE equivalent of  \/IR, 4
77  ( ps  \/  ch )  ( ph  ->  ( ps  \/  ch ) )  \/E 1,3,6 mpjaodan 764, 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 449 . . 3  |-  ( (
ph  /\  ps )  ->  ps )
21orcd 383 . 2  |-  ( (
ph  /\  ps )  ->  ( ps  \/  ch ) )
3 simpr 449 . . . 4  |-  ( (
ph  /\  ( ch  /\ 
th ) )  -> 
( ch  /\  th ) )
43simpld 447 . . 3  |-  ( (
ph  /\  ( ch  /\ 
th ) )  ->  ch )
54olcd 384 . 2  |-  ( (
ph  /\  ( ch  /\ 
th ) )  -> 
( ps  \/  ch ) )
6 ex-natded5.7.1 . 2  |-  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) )
72, 5, 6mpjaodan 764 1  |-  ( ph  ->  ( ps  \/  ch ) )
Colors of variables: wff set class
Syntax hints:    -> 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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