Description: Theorem 9.26 of [Clemente] p. 45, translated line by line using an
interpretation of natural deduction in Metamath. This proof has some
additional complications due to the fact that Metamath's existential
elimination rule does not change bound variables, so we need to verify
that 𝑥 is bound in the conclusion.
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 | 3 | ∃𝑥∀𝑦𝜓(𝑥, 𝑦) |
(𝜑 → ∃𝑥∀𝑦𝜓) |
Given |
$e. |
| 2 | 6 | ...| ∀𝑦𝜓(𝑥, 𝑦) |
((𝜑 ∧ ∀𝑦𝜓) → ∀𝑦𝜓) |
ND hypothesis assumption |
simpr 485. Later statements will have this scope. |
| 3 | 7;5,4 | ... 𝜓(𝑥, 𝑦) |
((𝜑 ∧ ∀𝑦𝜓) → 𝜓) |
∀E 2,y |
spsbcd 3725 (∀E), 5,6. To use it we need a1i 11 and vex 3443.
This could be immediately done with 19.21bi 2154, but we want to show
the general approach for substitution.
|
| 4 | 12;8,9,10,11 | ... ∃𝑥𝜓(𝑥, 𝑦) |
((𝜑 ∧ ∀𝑦𝜓) → ∃𝑥𝜓) |
∃I 3,a |
spesbcd 3800 (∃I), 11.
To use it we need sylibr 235, which in turn requires sylib 219 and
two uses of sbcid 3728.
This could be more immediately done using 19.8a 2146, but we want to show
the general approach for substitution.
|
| 5 | 13;1,2 | ∃𝑥𝜓(𝑥, 𝑦) |
(𝜑 → ∃𝑥𝜓) | ∃E 1,2,4,a |
exlimdd 2187 (∃E), 1,2,3,12.
We'll need supporting
assertions that the variable is free (not bound),
as provided in nfv 1896 and nfe1 2122 (MPE# 1,2) |
| 6 | 14 | ∀𝑦∃𝑥𝜓(𝑥, 𝑦) |
(𝜑 → ∀𝑦∃𝑥𝜓) |
∀I 5 |
alrimiv 1909 (∀I), 13 |
The original used Latin letters for predicates;
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.
Below is the final Metamath proof (which reorders some steps).
Note that in the original proof, 𝜓(𝑥, 𝑦) has explicit
parameters. In Metamath, these parameters are always implicit, and the
parameters upon which a wff variable can depend are recorded in the
"allowed substitution hints" below.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in ex-natded9.26-2 27887.
(Contributed by Mario Carneiro, 9-Feb-2017.)
(Revised by David A. Wheeler, 18-Feb-2017.)
(Proof modification is discouraged.) (New usage is
discouraged.) |
