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 |
((𝜓 ∧ 𝜒) → ¬ 𝜃) |
(𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃)) |
Given |
$e; adantr 474 to move it into the ND hypothesis |
2 | 3;4 | (𝜏 → 𝜃) |
(𝜑 → (𝜏 → 𝜃)) | Given |
$e; adantr 474 to move it into the ND hypothesis |
3 | 7;8 |
𝜒 | (𝜑 → 𝜒) |
Given |
$e; adantr 474 to move it into the ND hypothesis |
4 | 1;2 | 𝜏 | (𝜑 → 𝜏) |
Given |
$e. adantr 474 to move it into the ND hypothesis |
5 | 6 | ...| 𝜓 |
((𝜑 ∧ 𝜓) → 𝜓) |
ND Hypothesis/Assumption |
simpr 479. New ND hypothesis scope, each reference outside
the scope must change antecedent 𝜑 to (𝜑 ∧ 𝜓). |
6 | 9 | ... (𝜓 ∧ 𝜒) |
((𝜑 ∧ 𝜓) → (𝜓 ∧ 𝜒)) |
∧I 5,3 |
jca 507 (∧I), 6,8 (adantr 474 to bring in scope) |
7 | 5 | ... ¬ 𝜃 |
((𝜑 ∧ 𝜓) → ¬ 𝜃) |
→E 1,6 |
mpd 15 (→E), 2,4 |
8 | 12 | ... 𝜃 |
((𝜑 ∧ 𝜓) → 𝜃) |
→E 2,4 |
mpd 15 (→E), 9,11;
note the contradiction with ND line 7 (MPE line 5) |
9 | 13 | ¬ 𝜓 |
(𝜑 → ¬ 𝜓) |
¬I 5,7,8 |
pm2.65da 851 (¬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 𝜑 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 474; simpr 479 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 27825.
(Contributed by Mario Carneiro, 9-Feb-2017.)
(Proof modification is discouraged.) (New usage is
discouraged.) |