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Theorem ex-natded9.20 27601
Description: Theorem 9.20 of [Clemente] p. 43, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. 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
11 (𝜓 ∧ (𝜒𝜃)) (𝜑 → (𝜓 ∧ (𝜒𝜃))) Given $e
22 𝜓 (𝜑𝜓) EL 1 simpld 484 1
311 (𝜒𝜃) (𝜑 → (𝜒𝜃)) ER 1 simprd 485 1
44 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 473
55 ... (𝜓𝜒) ((𝜑𝜒) → (𝜓𝜒)) I 2,4 jca 503 3,4
66 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃))) IR 5 orcd 891 5
78 ...| 𝜃 ((𝜑𝜃) → 𝜃) ND hypothesis assumption simpr 473
89 ... (𝜓𝜃) ((𝜑𝜃) → (𝜓𝜃)) I 2,7 jca 503 7,8
910 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃))) IL 8 olcd 892 9
1012 ((𝜓𝜒) ∨ (𝜓𝜃)) (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃))) E 3,6,9 mpjaodan 972 6,10,11

The original used Latin letters; 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. To add an assumption, the antecedent is modified to include it (typically by using adantr 468; simpr 473 is useful when you want to depend directly on the new assumption). Below is the final Metamath proof (which reorders some steps).

A much more efficient proof is ex-natded9.20-2 27602. (Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis
Ref Expression
ex-natded9.20.1 (𝜑 → (𝜓 ∧ (𝜒𝜃)))
Assertion
Ref Expression
ex-natded9.20 (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))

Proof of Theorem ex-natded9.20
StepHypRef Expression
1 ex-natded9.20.1 . . . . . 6 (𝜑 → (𝜓 ∧ (𝜒𝜃)))
21simpld 484 . . . . 5 (𝜑𝜓)
32adantr 468 . . . 4 ((𝜑𝜒) → 𝜓)
4 simpr 473 . . . 4 ((𝜑𝜒) → 𝜒)
53, 4jca 503 . . 3 ((𝜑𝜒) → (𝜓𝜒))
65orcd 891 . 2 ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃)))
72adantr 468 . . . 4 ((𝜑𝜃) → 𝜓)
8 simpr 473 . . . 4 ((𝜑𝜃) → 𝜃)
97, 8jca 503 . . 3 ((𝜑𝜃) → (𝜓𝜃))
109olcd 892 . 2 ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃)))
111simprd 485 . 2 (𝜑 → (𝜒𝜃))
126, 10, 11mpjaodan 972 1 (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wo 865
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 198  df-an 385  df-or 866
This theorem is referenced by: (None)
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