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Theorem ex-natded5.7 21639
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 21640. 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 Given \$e. No need for adantr 452 because we do not move this into an ND hypothesis
21 ...| ND hypothesis assumption (new scope) simpr 448
32 ... IL 2 orcd 382, the MPE equivalent of IL, 1
43 ...| ND hypothesis assumption (new scope) simpr 448
54 ... EL 4 simpld 446, the MPE equivalent of EL, 3
66 ... IR 5 olcd 383, the MPE equivalent of IR, 4
77 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 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
Assertion
Ref Expression
ex-natded5.7

Proof of Theorem ex-natded5.7
StepHypRef Expression
1 simpr 448 . . 3
21orcd 382 . 2
3 simpr 448 . . . 4
43simpld 446 . . 3
54olcd 383 . 2
6 ex-natded5.7.1 . 2
72, 5, 6mpjaodan 762 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 358   wa 359 This theorem is referenced by:  ex-natded5.7-2  21640 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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