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Theorem ex-natded9.26 29669
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 3791 (E), 5,6. To use it we need a1i 11 and vex 3478. This could be immediately done with 19.21bi 2182, 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 3877 (I), 11. To use it we need sylibr 233, which in turn requires sylib 217 and two uses of sbcid 3794. This could be more immediately done using 19.8a 2174, but we want to show the general approach for substitution.
513;1,2 𝑥𝜓(𝑥, 𝑦) (𝜑 → ∃𝑥𝜓) E 1,2,4,a exlimdd 2213 (E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1917 and nfe1 2147 (MPE# 1,2)
614 𝑦𝑥𝜓(𝑥, 𝑦) (𝜑 → ∀𝑦𝑥𝜓) I 5 alrimiv 1930 (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 29670.

(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 1917 . . 3 𝑥𝜑
2 nfe1 2147 . . 3 𝑥𝑥𝜓
3 ex-natded9.26.1 . . 3 (𝜑 → ∃𝑥𝑦𝜓)
4 vex 3478 . . . . . . . 8 𝑦 ∈ V
54a1i 11 . . . . . . 7 ((𝜑 ∧ ∀𝑦𝜓) → 𝑦 ∈ V)
6 simpr 485 . . . . . . 7 ((𝜑 ∧ ∀𝑦𝜓) → ∀𝑦𝜓)
75, 6spsbcd 3791 . . . . . 6 ((𝜑 ∧ ∀𝑦𝜓) → [𝑦 / 𝑦]𝜓)
8 sbcid 3794 . . . . . 6 ([𝑦 / 𝑦]𝜓𝜓)
97, 8sylib 217 . . . . 5 ((𝜑 ∧ ∀𝑦𝜓) → 𝜓)
10 sbcid 3794 . . . . 5 ([𝑥 / 𝑥]𝜓𝜓)
119, 10sylibr 233 . . . 4 ((𝜑 ∧ ∀𝑦𝜓) → [𝑥 / 𝑥]𝜓)
1211spesbcd 3877 . . 3 ((𝜑 ∧ ∀𝑦𝜓) → ∃𝑥𝜓)
131, 2, 3, 12exlimdd 2213 . 2 (𝜑 → ∃𝑥𝜓)
1413alrimiv 1930 1 (𝜑 → ∀𝑦𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539  wex 1781  wcel 2106  Vcvv 3474  [wsbc 3777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-v 3476  df-sbc 3778
This theorem is referenced by: (None)
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