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Theorem ex-natded9.20 27258
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 475 1
311 (𝜒𝜃) (𝜑 → (𝜒𝜃)) ER 1 simprd 479 1
44 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 477
55 ... (𝜓𝜒) ((𝜑𝜒) → (𝜓𝜒)) I 2,4 jca 554 3,4
66 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃))) IR 5 orcd 407 5
78 ...| 𝜃 ((𝜑𝜃) → 𝜃) ND hypothesis assumption simpr 477
89 ... (𝜓𝜃) ((𝜑𝜃) → (𝜓𝜃)) I 2,7 jca 554 7,8
910 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃))) IL 8 olcd 408 9
1012 ((𝜓𝜒) ∨ (𝜓𝜃)) (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃))) E 3,6,9 mpjaodan 827 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 481; simpr 477 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 27259. (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 475 . . . . 5 (𝜑𝜓)
32adantr 481 . . . 4 ((𝜑𝜒) → 𝜓)
4 simpr 477 . . . 4 ((𝜑𝜒) → 𝜒)
53, 4jca 554 . . 3 ((𝜑𝜒) → (𝜓𝜒))
65orcd 407 . 2 ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃)))
72adantr 481 . . . 4 ((𝜑𝜃) → 𝜓)
8 simpr 477 . . . 4 ((𝜑𝜃) → 𝜃)
97, 8jca 554 . . 3 ((𝜑𝜃) → (𝜓𝜃))
109olcd 408 . 2 ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃)))
111simprd 479 . 2 (𝜑 → (𝜒𝜃))
126, 10, 11mpjaodan 827 1 (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384 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 197  df-or 385  df-an 386 This theorem is referenced by: (None)
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