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Theorem ex-natded9.26 28125
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 TranslationND Rationale MPE Rationale
13 𝑥𝑦𝜓(𝑥, 𝑦) (𝜑 → ∃𝑥𝑦𝜓) Given $e.
26 ...| 𝑦𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → ∀𝑦𝜓) ND hypothesis assumption simpr 485. Later statements will have this scope.
37;5,4 ... 𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → 𝜓) E 2,y spsbcd 3783 (E), 5,6. To use it we need a1i 11 and vex 3495. This could be immediately done with 19.21bi 2178, but we want to show the general approach for substitution.
412;8,9,10,11 ... 𝑥𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → ∃𝑥𝜓) I 3,a spesbcd 3863 (I), 11. To use it we need sylibr 235, which in turn requires sylib 219 and two uses of sbcid 3786. This could be more immediately done using 19.8a 2170, but we want to show the general approach for substitution.
513;1,2 𝑥𝜓(𝑥, 𝑦) (𝜑 → ∃𝑥𝜓) E 1,2,4,a exlimdd 2210 (E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1906 and nfe1 2145 (MPE# 1,2)
614 𝑦𝑥𝜓(𝑥, 𝑦) (𝜑 → ∀𝑦𝑥𝜓) I 5 alrimiv 1919 (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 28126.

(Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by David A. Wheeler, 18-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis
Ref Expression
ex-natded9.26.1 (𝜑 → ∃𝑥𝑦𝜓)
Assertion
Ref Expression
ex-natded9.26 (𝜑 → ∀𝑦𝑥𝜓)
Distinct variable group:   𝑥,𝑦,𝜑
Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem ex-natded9.26
StepHypRef Expression
1 nfv 1906 . . 3 𝑥𝜑
2 nfe1 2145 . . 3 𝑥𝑥𝜓
3 ex-natded9.26.1 . . 3 (𝜑 → ∃𝑥𝑦𝜓)
4 vex 3495 . . . . . . . 8 𝑦 ∈ V
54a1i 11 . . . . . . 7 ((𝜑 ∧ ∀𝑦𝜓) → 𝑦 ∈ V)
6 simpr 485 . . . . . . 7 ((𝜑 ∧ ∀𝑦𝜓) → ∀𝑦𝜓)
75, 6spsbcd 3783 . . . . . 6 ((𝜑 ∧ ∀𝑦𝜓) → [𝑦 / 𝑦]𝜓)
8 sbcid 3786 . . . . . 6 ([𝑦 / 𝑦]𝜓𝜓)
97, 8sylib 219 . . . . 5 ((𝜑 ∧ ∀𝑦𝜓) → 𝜓)
10 sbcid 3786 . . . . 5 ([𝑥 / 𝑥]𝜓𝜓)
119, 10sylibr 235 . . . 4 ((𝜑 ∧ ∀𝑦𝜓) → [𝑥 / 𝑥]𝜓)
1211spesbcd 3863 . . 3 ((𝜑 ∧ ∀𝑦𝜓) → ∃𝑥𝜓)
131, 2, 3, 12exlimdd 2210 . 2 (𝜑 → ∃𝑥𝜓)
1413alrimiv 1919 1 (𝜑 → ∀𝑦𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1526  wex 1771  wcel 2105  Vcvv 3492  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-sbc 3770
This theorem is referenced by: (None)
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