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Theorem ex-natded5.3 26422
Description: Theorem 5.3 of [Clemente] p. 16, translated line by line using an interpretation of natural deduction in Metamath. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.3-2 26423. A proof without context is shown in ex-natded5.3i 26424. 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:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3 (𝜓𝜒) (𝜑 → (𝜓𝜒)) Given $e; adantr 479 to move it into the ND hypothesis
25;6 (𝜒𝜃) (𝜑 → (𝜒𝜃)) Given $e; adantr 479 to move it into the ND hypothesis
31 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 475, to access the new assumption
44 ... 𝜒 ((𝜑𝜓) → 𝜒) E 1,3 mpd 15, the MPE equivalent of E, 1.3. adantr 479 was used to transform its dependency (we could also use imp 443 to get this directly from 1)
57 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15, the MPE equivalent of E, 4,6. adantr 479 was used to transform its dependency
68 ... (𝜒𝜃) ((𝜑𝜓) → (𝜒𝜃)) I 4,5 jca 552, the MPE equivalent of I, 4,7
79 (𝜓 → (𝜒𝜃)) (𝜑 → (𝜓 → (𝜒𝜃))) I 3,6 ex 448, the MPE equivalent of I, 8

The original used Latin letters for predicates; 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. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses
Ref Expression
ex-natded5.3.1 (𝜑 → (𝜓𝜒))
ex-natded5.3.2 (𝜑 → (𝜒𝜃))
Assertion
Ref Expression
ex-natded5.3 (𝜑 → (𝜓 → (𝜒𝜃)))

Proof of Theorem ex-natded5.3
StepHypRef Expression
1 simpr 475 . . . 4 ((𝜑𝜓) → 𝜓)
2 ex-natded5.3.1 . . . . 5 (𝜑 → (𝜓𝜒))
32adantr 479 . . . 4 ((𝜑𝜓) → (𝜓𝜒))
41, 3mpd 15 . . 3 ((𝜑𝜓) → 𝜒)
5 ex-natded5.3.2 . . . . 5 (𝜑 → (𝜒𝜃))
65adantr 479 . . . 4 ((𝜑𝜓) → (𝜒𝜃))
74, 6mpd 15 . . 3 ((𝜑𝜓) → 𝜃)
84, 7jca 552 . 2 ((𝜑𝜓) → (𝜒𝜃))
98ex 448 1 (𝜑 → (𝜓 → (𝜒𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382
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 195  df-an 384
This theorem is referenced by: (None)
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