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Theorem ex-natded5.7 27971
Description: Theorem 5.7 of [Clemente] p. 19, translated line by line using the interpretation of natural deduction in Metamath. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.7-2 27972. 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
16 (𝜓 ∨ (𝜒𝜃)) (𝜑 → (𝜓 ∨ (𝜒𝜃))) Given \$e. No need for adantr 473 because we do not move this into an ND hypothesis
21 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption (new scope) simpr 477
32 ... (𝜓𝜒) ((𝜑𝜓) → (𝜓𝜒)) IL 2 orcd 859, the MPE equivalent of IL, 1
43 ...| (𝜒𝜃) ((𝜑 ∧ (𝜒𝜃)) → (𝜒𝜃)) ND hypothesis assumption (new scope) simpr 477
54 ... 𝜒 ((𝜑 ∧ (𝜒𝜃)) → 𝜒) EL 4 simpld 487, the MPE equivalent of EL, 3
66 ... (𝜓𝜒) ((𝜑 ∧ (𝜒𝜃)) → (𝜓𝜒)) IR 5 olcd 860, the MPE equivalent of IR, 4
77 (𝜓𝜒) (𝜑 → (𝜓𝜒)) E 1,3,6 mpjaodan 941, the MPE equivalent of E, 2,5,6

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. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis
Ref Expression
ex-natded5.7.1 (𝜑 → (𝜓 ∨ (𝜒𝜃)))
Assertion
Ref Expression
ex-natded5.7 (𝜑 → (𝜓𝜒))

Proof of Theorem ex-natded5.7
StepHypRef Expression
1 simpr 477 . . 3 ((𝜑𝜓) → 𝜓)
21orcd 859 . 2 ((𝜑𝜓) → (𝜓𝜒))
3 simpr 477 . . . 4 ((𝜑 ∧ (𝜒𝜃)) → (𝜒𝜃))
43simpld 487 . . 3 ((𝜑 ∧ (𝜒𝜃)) → 𝜒)
54olcd 860 . 2 ((𝜑 ∧ (𝜒𝜃)) → (𝜓𝜒))
6 ex-natded5.7.1 . 2 (𝜑 → (𝜓 ∨ (𝜒𝜃)))
72, 5, 6mpjaodan 941 1 (𝜑 → (𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   ∨ wo 833 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 199  df-an 388  df-or 834 This theorem is referenced by: (None)
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