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 Translation  ND Rationale 
MPE Rationale 
1  10;11 


Given 
$e; adantr 451 to move it into the ND hypothesis 
2  3;4  
 Given 
$e; adantr 451 to move it into the ND hypothesis 
3  7;8 
 
Given 
$e; adantr 451 to move it into the ND hypothesis 
4  1;2   
Given 
$e. adantr 451 to move it into the ND hypothesis 
5  6  ... 

ND Hypothesis/Assumption 
simpr 447. New ND hypothesis scope, each reference outside
the scope must change antedent to . 
6  9  ... 

I 5,3 
jca 518 (I), 6,8 (adantr 451 to bring in scope) 
7  5  ... 

E 1,6 
mpd 14 (E), 2,4 
8  12  ... 

E 2,4 
mpd 14 (E), 9,11;
note the contradiction with ND line 7 (MPE line 5) 
9  13  

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 lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 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 exnatded5.82 20817.
(Proof modification is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.) 