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Theorem ex-natded5.2 26402
Description: Theorem 5.2 of [Clemente] p. 15, translated line by line using the interpretation of natural deduction in Metamath. 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:
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
15 ((𝜓𝜒) → 𝜃) (𝜑 → ((𝜓𝜒) → 𝜃)) Given \$e.
22 (𝜒𝜓) (𝜑 → (𝜒𝜓)) Given \$e.
31 𝜒 (𝜑𝜒) Given \$e.
43 𝜓 (𝜑𝜓) E 2,3 mpd 15, the MPE equivalent of E, 1,2
54 (𝜓𝜒) (𝜑 → (𝜓𝜒)) I 4,3 jca 552, the MPE equivalent of I, 3,1
66 𝜃 (𝜑𝜃) E 1,5 mpd 15, the MPE equivalent of E, 4,5

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). A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.2-2 26403. A proof without context is shown in ex-natded5.2i 26404. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses
Ref Expression
ex-natded5.2.1 (𝜑 → ((𝜓𝜒) → 𝜃))
ex-natded5.2.2 (𝜑 → (𝜒𝜓))
ex-natded5.2.3 (𝜑𝜒)
Assertion
Ref Expression
ex-natded5.2 (𝜑𝜃)

Proof of Theorem ex-natded5.2
StepHypRef Expression
1 ex-natded5.2.3 . . . 4 (𝜑𝜒)
2 ex-natded5.2.2 . . . 4 (𝜑 → (𝜒𝜓))
31, 2mpd 15 . . 3 (𝜑𝜓)
43, 1jca 552 . 2 (𝜑 → (𝜓𝜒))
5 ex-natded5.2.1 . 2 (𝜑 → ((𝜓𝜒) → 𝜃))
64, 5mpd 15 1 (𝜑𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 195  df-an 384 This theorem is referenced by: (None)
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