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Theorem ex-natded5.8 27141
Description: Theorem 5.8 of [Clemente] p. 20, 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
110;11 ((𝜓𝜒) → ¬ 𝜃) (𝜑 → ((𝜓𝜒) → ¬ 𝜃)) Given \$e; adantr 481 to move it into the ND hypothesis
23;4 (𝜏𝜃) (𝜑 → (𝜏𝜃)) Given \$e; adantr 481 to move it into the ND hypothesis
37;8 𝜒 (𝜑𝜒) Given \$e; adantr 481 to move it into the ND hypothesis
41;2 𝜏 (𝜑𝜏) Given \$e. adantr 481 to move it into the ND hypothesis
56 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND Hypothesis/Assumption simpr 477. New ND hypothesis scope, each reference outside the scope must change antecedent 𝜑 to (𝜑𝜓).
69 ... (𝜓𝜒) ((𝜑𝜓) → (𝜓𝜒)) I 5,3 jca 554 (I), 6,8 (adantr 481 to bring in scope)
75 ... ¬ 𝜃 ((𝜑𝜓) → ¬ 𝜃) E 1,6 mpd 15 (E), 2,4
812 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15 (E), 9,11; note the contradiction with ND line 7 (MPE line 5)
913 ¬ 𝜓 (𝜑 → ¬ 𝜓) ¬I 5,7,8 pm2.65da 599 (¬I), 5,12; proof by contradiction. MPE step 6 (ND#5) does not need a reference here, because the assumption is embedded in the antecedents

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, using more of Metamath and MPE's capabilities, is shown in ex-natded5.8-2 27142.

(Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses
Ref Expression
ex-natded5.8.1 (𝜑 → ((𝜓𝜒) → ¬ 𝜃))
ex-natded5.8.2 (𝜑 → (𝜏𝜃))
ex-natded5.8.3 (𝜑𝜒)
ex-natded5.8.4 (𝜑𝜏)
Assertion
Ref Expression
ex-natded5.8 (𝜑 → ¬ 𝜓)

Proof of Theorem ex-natded5.8
StepHypRef Expression
1 ex-natded5.8.4 . . . 4 (𝜑𝜏)
21adantr 481 . . 3 ((𝜑𝜓) → 𝜏)
3 ex-natded5.8.2 . . . 4 (𝜑 → (𝜏𝜃))
43adantr 481 . . 3 ((𝜑𝜓) → (𝜏𝜃))
52, 4mpd 15 . 2 ((𝜑𝜓) → 𝜃)
6 simpr 477 . . . 4 ((𝜑𝜓) → 𝜓)
7 ex-natded5.8.3 . . . . 5 (𝜑𝜒)
87adantr 481 . . . 4 ((𝜑𝜓) → 𝜒)
96, 8jca 554 . . 3 ((𝜑𝜓) → (𝜓𝜒))
10 ex-natded5.8.1 . . . 4 (𝜑 → ((𝜓𝜒) → ¬ 𝜃))
1110adantr 481 . . 3 ((𝜑𝜓) → ((𝜓𝜒) → ¬ 𝜃))
129, 11mpd 15 . 2 ((𝜑𝜓) → ¬ 𝜃)
135, 12pm2.65da 599 1 (𝜑 → ¬ 𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ 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-an 386 This theorem is referenced by: (None)
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