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Theorem ex-natded9.20 27888
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 495 1
311 (𝜒𝜃) (𝜑 → (𝜒𝜃)) ER 1 simprd 496 1
44 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 485
55 ... (𝜓𝜒) ((𝜑𝜒) → (𝜓𝜒)) I 2,4 jca 512 3,4
66 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃))) IR 5 orcd 870 5
78 ...| 𝜃 ((𝜑𝜃) → 𝜃) ND hypothesis assumption simpr 485
89 ... (𝜓𝜃) ((𝜑𝜃) → (𝜓𝜃)) I 2,7 jca 512 7,8
910 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃))) IL 8 olcd 871 9
1012 ((𝜓𝜒) ∨ (𝜓𝜃)) (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃))) E 3,6,9 mpjaodan 953 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 485 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 27889. (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 495 . . . . 5 (𝜑𝜓)
32adantr 481 . . . 4 ((𝜑𝜒) → 𝜓)
4 simpr 485 . . . 4 ((𝜑𝜒) → 𝜒)
53, 4jca 512 . . 3 ((𝜑𝜒) → (𝜓𝜒))
65orcd 870 . 2 ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃)))
72adantr 481 . . . 4 ((𝜑𝜃) → 𝜓)
8 simpr 485 . . . 4 ((𝜑𝜃) → 𝜃)
97, 8jca 512 . . 3 ((𝜑𝜃) → (𝜓𝜃))
109olcd 871 . 2 ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃)))
111simprd 496 . 2 (𝜑 → (𝜒𝜃))
126, 10, 11mpjaodan 953 1 (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 842 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 208  df-an 397  df-or 843 This theorem is referenced by: (None)
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