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Theorem ex-natded9.20 28367
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 498 1
311 (𝜒𝜃) (𝜑 → (𝜒𝜃)) ER 1 simprd 499 1
44 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 488
55 ... (𝜓𝜒) ((𝜑𝜒) → (𝜓𝜒)) I 2,4 jca 515 3,4
66 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃))) IR 5 orcd 872 5
78 ...| 𝜃 ((𝜑𝜃) → 𝜃) ND hypothesis assumption simpr 488
89 ... (𝜓𝜃) ((𝜑𝜃) → (𝜓𝜃)) I 2,7 jca 515 7,8
910 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃))) IL 8 olcd 873 9
1012 ((𝜓𝜒) ∨ (𝜓𝜃)) (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃))) E 3,6,9 mpjaodan 958 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 484; simpr 488 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 28368. (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 498 . . . . 5 (𝜑𝜓)
32adantr 484 . . . 4 ((𝜑𝜒) → 𝜓)
4 simpr 488 . . . 4 ((𝜑𝜒) → 𝜒)
53, 4jca 515 . . 3 ((𝜑𝜒) → (𝜓𝜒))
65orcd 872 . 2 ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃)))
72adantr 484 . . . 4 ((𝜑𝜃) → 𝜓)
8 simpr 488 . . . 4 ((𝜑𝜃) → 𝜃)
97, 8jca 515 . . 3 ((𝜑𝜃) → (𝜓𝜃))
109olcd 873 . 2 ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃)))
111simprd 499 . 2 (𝜑 → (𝜒𝜃))
126, 10, 11mpjaodan 958 1 (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 846
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 210  df-an 400  df-or 847
This theorem is referenced by: (None)
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