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Theorem ex-natded5.2 20791
Description: Theorem 5.2 of [Clemente] p. 15, 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. The original proof, which uses Fitch style, was written as follows:
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
15  ( ( ps  /\  ch )  ->  th )  ( ph  ->  ( ( ps  /\  ch )  ->  th ) ) Given $e.
22  ( ch  ->  ps )  ( ph  ->  ( ch  ->  ps ) ) Given $e.
31  ch  ( ph  ->  ch ) Given $e.
43  ps  ( ph  ->  ps )  ->E 2,3 mpd 14, the MPE equivalent of  ->E, 1,2
54  ( ps  /\  ch )  ( ph  ->  ( ps  /\  ch ) )  /\I 4,3 jca 518, the MPE equivalent of  /\I, 3,1
66  th  ( ph  ->  th )  ->E 1,5 mpd 14, the MPE equivalent of  ->E, 4,5

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. Below is the final metamath proof (which reorders some steps). A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.2-2 20792. A proof without context is shown in ex-natded5.2i 20793. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Hypotheses
Ref Expression
ex-natded5.2.1  |-  ( ph  ->  ( ( ps  /\  ch )  ->  th )
)
ex-natded5.2.2  |-  ( ph  ->  ( ch  ->  ps ) )
ex-natded5.2.3  |-  ( ph  ->  ch )
Assertion
Ref Expression
ex-natded5.2  |-  ( ph  ->  th )

Proof of Theorem ex-natded5.2
StepHypRef Expression
1 ex-natded5.2.3 . . . 4  |-  ( ph  ->  ch )
2 ex-natded5.2.2 . . . 4  |-  ( ph  ->  ( ch  ->  ps ) )
31, 2mpd 14 . . 3  |-  ( ph  ->  ps )
43, 1jca 518 . 2  |-  ( ph  ->  ( ps  /\  ch ) )
5 ex-natded5.2.1 . 2  |-  ( ph  ->  ( ( ps  /\  ch )  ->  th )
)
64, 5mpd 14 1  |-  ( ph  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ 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-an 360
  Copyright terms: Public domain W3C validator