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Theorem ex-natded9.26 29936
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 484. Later statements will have this scope.
37;5,4 ... 𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → 𝜓) E 2,y spsbcd 3792 (E), 5,6. To use it we need a1i 11 and vex 3477. This could be immediately done with 19.21bi 2181, 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 3878 (I), 11. To use it we need sylibr 233, which in turn requires sylib 217 and two uses of sbcid 3795. This could be more immediately done using 19.8a 2173, but we want to show the general approach for substitution.
513;1,2 𝑥𝜓(𝑥, 𝑦) (𝜑 → ∃𝑥𝜓) E 1,2,4,a exlimdd 2212 (E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1916 and nfe1 2146 (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 29937.

(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 2146 . . 3 𝑥𝑥𝜓
3 ex-natded9.26.1 . . 3 (𝜑 → ∃𝑥𝑦𝜓)
4 vex 3477 . . . . . . . 8 𝑦 ∈ V
54a1i 11 . . . . . . 7 ((𝜑 ∧ ∀𝑦𝜓) → 𝑦 ∈ V)
6 simpr 484 . . . . . . 7 ((𝜑 ∧ ∀𝑦𝜓) → ∀𝑦𝜓)
75, 6spsbcd 3792 . . . . . 6 ((𝜑 ∧ ∀𝑦𝜓) → [𝑦 / 𝑦]𝜓)
8 sbcid 3795 . . . . . 6 ([𝑦 / 𝑦]𝜓𝜓)
97, 8sylib 217 . . . . 5 ((𝜑 ∧ ∀𝑦𝜓) → 𝜓)
10 sbcid 3795 . . . . 5 ([𝑥 / 𝑥]𝜓𝜓)
119, 10sylibr 233 . . . 4 ((𝜑 ∧ ∀𝑦𝜓) → [𝑥 / 𝑥]𝜓)
1211spesbcd 3878 . . 3 ((𝜑 ∧ ∀𝑦𝜓) → ∃𝑥𝜓)
131, 2, 3, 12exlimdd 2212 . 2 (𝜑 → ∃𝑥𝜓)
1413alrimiv 1929 1 (𝜑 → ∀𝑦𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538  wex 1780  wcel 2105  Vcvv 3473  [wsbc 3778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-v 3475  df-sbc 3779
This theorem is referenced by: (None)
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