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Theorem ex-natded5.8 20822
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 453 to move it into the ND hypothesis
23;4  ( ta  ->  th )  ( ph  ->  ( ta  ->  th ) ) Given $e; adantr 453 to move it into the ND hypothesis
37;8  ch  ( ph  ->  ch ) Given $e; adantr 453 to move it into the ND hypothesis
41;2  ta  ( ph  ->  ta ) Given $e. adantr 453 to move it into the ND hypothesis
56 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND Hypothesis/Assumption simpr 449. 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 520 ( /\I), 6,8 (adantr 453 to bring in scope)
75 ...  -.  th  ( ( ph  /\  ps )  ->  -.  th )  ->E 1,6 mpd 16 ( ->E), 2,4
812 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 16 ( ->E), 9,11; note the contradiction with ND line 7 (MPE line 5)
913  -.  ps  ( ph  ->  -.  ps )  -.I 5,7,8 pm2.65da 561 ( -.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 453; simpr 449 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 20823.

(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 453 . . 3  |-  ( (
ph  /\  ps )  ->  ta )
3 ex-natded5.8.2 . . . 4  |-  ( ph  ->  ( ta  ->  th )
)
43adantr 453 . . 3  |-  ( (
ph  /\  ps )  ->  ( ta  ->  th )
)
52, 4mpd 16 . 2  |-  ( (
ph  /\  ps )  ->  th )
6 simpr 449 . . . 4  |-  ( (
ph  /\  ps )  ->  ps )
7 ex-natded5.8.3 . . . . 5  |-  ( ph  ->  ch )
87adantr 453 . . . 4  |-  ( (
ph  /\  ps )  ->  ch )
96, 8jca 520 . . 3  |-  ( (
ph  /\  ps )  ->  ( ps  /\  ch ) )
10 ex-natded5.8.1 . . . 4  |-  ( ph  ->  ( ( ps  /\  ch )  ->  -.  th ) )
1110adantr 453 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( ps  /\  ch )  ->  -.  th ) )
129, 11mpd 16 . 2  |-  ( (
ph  /\  ps )  ->  -.  th )
135, 12pm2.65da 561 1  |-  ( ph  ->  -.  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ 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-an 362
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