Description: Theorem 9.20 of [Clemente] p. 43, 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 | 1 |
(𝜓 ∧ (𝜒 ∨ 𝜃)) |
(𝜑 → (𝜓 ∧ (𝜒 ∨ 𝜃))) |
Given |
$e |
| 2 | 2 | 𝜓 |
(𝜑 → 𝜓) |
∧EL 1 |
simpld 499 1 |
| 3 | 11 |
(𝜒 ∨ 𝜃) |
(𝜑 → (𝜒 ∨ 𝜃)) |
∧ER 1 |
simprd 500 1 |
| 4 | 4 |
...| 𝜒 |
((𝜑 ∧ 𝜒) → 𝜒) |
ND hypothesis assumption |
simpr 489 |
| 5 | 5 |
... (𝜓 ∧ 𝜒) |
((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒)) |
∧I 2,4 |
jca 520 3,4 |
| 6 | 6 |
... ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) |
((𝜑 ∧ 𝜒) → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) |
∨IR 5 |
orcd 886 5 |
| 7 | 8 |
...| 𝜃 |
((𝜑 ∧ 𝜃) → 𝜃) |
ND hypothesis assumption |
simpr 489 |
| 8 | 9 |
... (𝜓 ∧ 𝜃) |
((𝜑 ∧ 𝜃) → (𝜓 ∧ 𝜃)) |
∧I 2,7 |
jca 520 7,8 |
| 9 | 10 |
... ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) |
((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) |
∨IL 8 |
olcd 887 9 |
| 10 | 12 |
((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) |
(𝜑 → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) |
∨E 3,6,9 |
mpjaodan 973 6,10,11 |
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 485; simpr 489 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 ex-natded9.20-2 30709.
(Contributed by David A. Wheeler, 19-Feb-2017.)
(Proof modification is discouraged.) (New usage is
discouraged.) |
