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Theorem ex-natded9.20 30266
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 493 1
311 (𝜒𝜃) (𝜑 → (𝜒𝜃)) ER 1 simprd 494 1
44 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 483
55 ... (𝜓𝜒) ((𝜑𝜒) → (𝜓𝜒)) I 2,4 jca 510 3,4
66 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃))) IR 5 orcd 871 5
78 ...| 𝜃 ((𝜑𝜃) → 𝜃) ND hypothesis assumption simpr 483
89 ... (𝜓𝜃) ((𝜑𝜃) → (𝜓𝜃)) I 2,7 jca 510 7,8
910 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃))) IL 8 olcd 872 9
1012 ((𝜓𝜒) ∨ (𝜓𝜃)) (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃))) E 3,6,9 mpjaodan 956 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 479; simpr 483 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 30267. (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 493 . . . . 5 (𝜑𝜓)
32adantr 479 . . . 4 ((𝜑𝜒) → 𝜓)
4 simpr 483 . . . 4 ((𝜑𝜒) → 𝜒)
53, 4jca 510 . . 3 ((𝜑𝜒) → (𝜓𝜒))
65orcd 871 . 2 ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃)))
72adantr 479 . . . 4 ((𝜑𝜃) → 𝜓)
8 simpr 483 . . . 4 ((𝜑𝜃) → 𝜃)
97, 8jca 510 . . 3 ((𝜑𝜃) → (𝜓𝜃))
109olcd 872 . 2 ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃)))
111simprd 494 . 2 (𝜑 → (𝜒𝜃))
126, 10, 11mpjaodan 956 1 (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845
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 206  df-an 395  df-or 846
This theorem is referenced by: (None)
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