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Theorem ex-natded9.26 28296
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 489. Later statements will have this scope.
37;5,4 ... 𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → 𝜓) E 2,y spsbcd 3711 (E), 5,6. To use it we need a1i 11 and vex 3414. This could be immediately done with 19.21bi 2187, 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 3790 (I), 11. To use it we need sylibr 237, which in turn requires sylib 221 and two uses of sbcid 3714. This could be more immediately done using 19.8a 2179, but we want to show the general approach for substitution.
513;1,2 𝑥𝜓(𝑥, 𝑦) (𝜑 → ∃𝑥𝜓) E 1,2,4,a exlimdd 2219 (E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1916 and nfe1 2152 (MPE# 1,2)
614 𝑦𝑥𝜓(𝑥, 𝑦) (𝜑 → ∀𝑦𝑥𝜓) I 5 alrimiv 1929 (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 28297.

(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 1916 . . 3 𝑥𝜑
2 nfe1 2152 . . 3 𝑥𝑥𝜓
3 ex-natded9.26.1 . . 3 (𝜑 → ∃𝑥𝑦𝜓)
4 vex 3414 . . . . . . . 8 𝑦 ∈ V
54a1i 11 . . . . . . 7 ((𝜑 ∧ ∀𝑦𝜓) → 𝑦 ∈ V)
6 simpr 489 . . . . . . 7 ((𝜑 ∧ ∀𝑦𝜓) → ∀𝑦𝜓)
75, 6spsbcd 3711 . . . . . 6 ((𝜑 ∧ ∀𝑦𝜓) → [𝑦 / 𝑦]𝜓)
8 sbcid 3714 . . . . . 6 ([𝑦 / 𝑦]𝜓𝜓)
97, 8sylib 221 . . . . 5 ((𝜑 ∧ ∀𝑦𝜓) → 𝜓)
10 sbcid 3714 . . . . 5 ([𝑥 / 𝑥]𝜓𝜓)
119, 10sylibr 237 . . . 4 ((𝜑 ∧ ∀𝑦𝜓) → [𝑥 / 𝑥]𝜓)
1211spesbcd 3790 . . 3 ((𝜑 ∧ ∀𝑦𝜓) → ∃𝑥𝜓)
131, 2, 3, 12exlimdd 2219 . 2 (𝜑 → ∃𝑥𝜓)
1413alrimiv 1929 1 (𝜑 → ∀𝑦𝑥𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1537  ∃wex 1782   ∈ wcel 2112  Vcvv 3410  [wsbc 3697 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-v 3412  df-sbc 3698 This theorem is referenced by: (None)
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