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Theorem ex-natded5.5 27979
Description: Theorem 5.5 of [Clemente] p. 18, 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
12;3 (𝜓𝜒) (𝜑 → (𝜓𝜒)) Given \$e; adantr 473 to move it into the ND hypothesis
25 ¬ 𝜒 (𝜑 → ¬ 𝜒) Given \$e; we'll use adantr 473 to move it into the ND hypothesis
31 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 477
44 ... 𝜒 ((𝜑𝜓) → 𝜒) E 1,3 mpd 15 1,3
56 ... ¬ 𝜒 ((𝜑𝜓) → ¬ 𝜒) IT 2 adantr 473 5
67 ¬ 𝜓 (𝜑 → ¬ 𝜓) I 3,4,5 pm2.65da 804 4,6

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 473; 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 mtod 190; a proof without context is shown in mto 189.

(Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses
Ref Expression
ex-natded5.5.1 (𝜑 → (𝜓𝜒))
ex-natded5.5.2 (𝜑 → ¬ 𝜒)
Assertion
Ref Expression
ex-natded5.5 (𝜑 → ¬ 𝜓)

Proof of Theorem ex-natded5.5
StepHypRef Expression
1 simpr 477 . . 3 ((𝜑𝜓) → 𝜓)
2 ex-natded5.5.1 . . . 4 (𝜑 → (𝜓𝜒))
32adantr 473 . . 3 ((𝜑𝜓) → (𝜓𝜒))
41, 3mpd 15 . 2 ((𝜑𝜓) → 𝜒)
5 ex-natded5.5.2 . . 3 (𝜑 → ¬ 𝜒)
65adantr 473 . 2 ((𝜑𝜓) → ¬ 𝜒)
74, 6pm2.65da 804 1 (𝜑 → ¬ 𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387 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 199  df-an 388 This theorem is referenced by: (None)
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