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Theorem ex-natded5.5 21568
Description: Theorem 5.5 of [Clemente] p. 18, 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
12;3  ( ps  ->  ch )  ( ph  ->  ( ps  ->  ch ) ) Given $e; adantr 452 to move it into the ND hypothesis
25  -.  ch  ( ph  ->  -.  ch ) Given $e; we'll use adantr 452 to move it into the ND hypothesis
31 ...|  ps  ( ph  ->  ps ) ND hypothesis assumption simpr 448
44 ...  ch  ( ( ph  /\  ps )  ->  ch )  ->E 1,3 mpd 15 1,3
56 ...  -.  ch  ( ( ph  /\  ps )  ->  -.  ch ) IT 2 adantr 452 5
67  -.  ps  ( ph  ->  -.  ps )  /\I 3,4,5 pm2.65da 560 4,6

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 452; simpr 448 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 is mtod 170; a proof without context is shown in mto 169.

(Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

Hypotheses
Ref Expression
ex-natded5.5.1  |-  ( ph  ->  ( ps  ->  ch ) )
ex-natded5.5.2  |-  ( ph  ->  -.  ch )
Assertion
Ref Expression
ex-natded5.5  |-  ( ph  ->  -.  ps )

Proof of Theorem ex-natded5.5
StepHypRef Expression
1 simpr 448 . . 3  |-  ( (
ph  /\  ps )  ->  ps )
2 ex-natded5.5.1 . . . 4  |-  ( ph  ->  ( ps  ->  ch ) )
32adantr 452 . . 3  |-  ( (
ph  /\  ps )  ->  ( ps  ->  ch ) )
41, 3mpd 15 . 2  |-  ( (
ph  /\  ps )  ->  ch )
5 ex-natded5.5.2 . . 3  |-  ( ph  ->  -.  ch )
65adantr 452 . 2  |-  ( (
ph  /\  ps )  ->  -.  ch )
74, 6pm2.65da 560 1  |-  ( ph  ->  -.  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359
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-an 361
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