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Theorem ex-natded5.8 20800
Description: Theorem 5.8 of [Clemente] p. 20, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. 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 (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
110;11  ( ( ps  /\  ch )  ->  -.  th )  ( ph  ->  ( ( ps  /\  ch )  ->  -.  th ) ) Given $e; adantr 451 to move it into the ND hypothesis
23;4  ( ta  ->  th )  ( ph  ->  ( ta  ->  th ) ) Given $e; adantr 451 to move it into the ND hypothesis
37;8  ch  ( ph  ->  ch ) Given $e; adantr 451 to move it into the ND hypothesis
41;2  ta  ( ph  ->  ta ) Given $e. adantr 451 to move it into the ND hypothesis
56 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND Hypothesis/Assumption simpr 447. New ND hypothesis scope, each reference outside the scope must change antedent  ph to  ( ph  /\  ps ).
69 ...  ( ps  /\  ch )  ( ( ph  /\  ps )  ->  ( ps  /\  ch ) )  /\I 5,3 jca 518 ( /\I), 6,8 (adantr 451 to bring in scope)
75 ...  -.  th  ( ( ph  /\  ps )  ->  -.  th )  ->E 1,6 mpd 14 ( ->E), 2,4
812 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 14 ( ->E), 9,11; note the contradiction with ND line 7 (MPE line 5)
913  -.  ps  ( ph  ->  -.  ps )  -.I 5,7,8 pm2.65da 559 ( -.I), 5,12; proof by contradiction. MPE step 6 (ND#5) does not need a reference here, because the assumption is embedded in the antecedents

The original used Latin letters; 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. To add an assumption, the antecedent is modified to include it (typically by using adantr 451; simpr 447 is useful when you want to depend directly on the new assumption). 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-natded5.8-2 20801.

(Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Hypotheses
Ref Expression
ex-natded5.8.1  |-  ( ph  ->  ( ( ps  /\  ch )  ->  -.  th ) )
ex-natded5.8.2  |-  ( ph  ->  ( ta  ->  th )
)
ex-natded5.8.3  |-  ( ph  ->  ch )
ex-natded5.8.4  |-  ( ph  ->  ta )
Assertion
Ref Expression
ex-natded5.8  |-  ( ph  ->  -.  ps )

Proof of Theorem ex-natded5.8
StepHypRef Expression
1 ex-natded5.8.4 . . . 4  |-  ( ph  ->  ta )
21adantr 451 . . 3  |-  ( (
ph  /\  ps )  ->  ta )
3 ex-natded5.8.2 . . . 4  |-  ( ph  ->  ( ta  ->  th )
)
43adantr 451 . . 3  |-  ( (
ph  /\  ps )  ->  ( ta  ->  th )
)
52, 4mpd 14 . 2  |-  ( (
ph  /\  ps )  ->  th )
6 simpr 447 . . . 4  |-  ( (
ph  /\  ps )  ->  ps )
7 ex-natded5.8.3 . . . . 5  |-  ( ph  ->  ch )
87adantr 451 . . . 4  |-  ( (
ph  /\  ps )  ->  ch )
96, 8jca 518 . . 3  |-  ( (
ph  /\  ps )  ->  ( ps  /\  ch ) )
10 ex-natded5.8.1 . . . 4  |-  ( ph  ->  ( ( ps  /\  ch )  ->  -.  th ) )
1110adantr 451 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( ps  /\  ch )  ->  -.  th ) )
129, 11mpd 14 . 2  |-  ( (
ph  /\  ps )  ->  -.  th )
135, 12pm2.65da 559 1  |-  ( ph  ->  -.  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> 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