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Theorem ex-natded5.13 29401
Description: Theorem 5.13 of [Clemente] p. 20, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.13-2 29402. 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
115 (𝜓𝜒) (𝜑 → (𝜓𝜒)) Given $e.
2;32 (𝜓𝜃) (𝜑 → (𝜓𝜃)) Given $e. adantr 482 to move it into the ND hypothesis
39 𝜏 → ¬ 𝜒) (𝜑 → (¬ 𝜏 → ¬ 𝜒)) Given $e. ad2antrr 725 to move it into the ND sub-hypothesis
41 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 486
54 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15 1,3
65 ... (𝜃𝜏) ((𝜑𝜓) → (𝜃𝜏)) I 5 orcd 872 4
76 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 486
88 ... ...| ¬ 𝜏 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜏) (sub) ND hypothesis assumption simpr 486
911 ... ... ¬ 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜒) E 3,8 mpd 15 8,10
107 ... ... 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → 𝜒) IT 7 adantr 482 6
1112 ... ¬ ¬ 𝜏 ((𝜑𝜒) → ¬ ¬ 𝜏) ¬I 8,9,10 pm2.65da 816 7,11
1213 ... 𝜏 ((𝜑𝜒) → 𝜏) ¬E 11 notnotrd 133 12
1314 ... (𝜃𝜏) ((𝜑𝜒) → (𝜃𝜏)) I 12 olcd 873 13
1416 (𝜃𝜏) (𝜑 → (𝜃𝜏)) E 1,6,13 mpjaodan 958 5,14,15

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 482; simpr 486 is useful when you want to depend directly on the new assumption). (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses
Ref Expression
ex-natded5.13.1 (𝜑 → (𝜓𝜒))
ex-natded5.13.2 (𝜑 → (𝜓𝜃))
ex-natded5.13.3 (𝜑 → (¬ 𝜏 → ¬ 𝜒))
Assertion
Ref Expression
ex-natded5.13 (𝜑 → (𝜃𝜏))

Proof of Theorem ex-natded5.13
StepHypRef Expression
1 simpr 486 . . . 4 ((𝜑𝜓) → 𝜓)
2 ex-natded5.13.2 . . . . 5 (𝜑 → (𝜓𝜃))
32adantr 482 . . . 4 ((𝜑𝜓) → (𝜓𝜃))
41, 3mpd 15 . . 3 ((𝜑𝜓) → 𝜃)
54orcd 872 . 2 ((𝜑𝜓) → (𝜃𝜏))
6 simpr 486 . . . . . 6 ((𝜑𝜒) → 𝜒)
76adantr 482 . . . . 5 (((𝜑𝜒) ∧ ¬ 𝜏) → 𝜒)
8 simpr 486 . . . . . 6 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜏)
9 ex-natded5.13.3 . . . . . . 7 (𝜑 → (¬ 𝜏 → ¬ 𝜒))
109ad2antrr 725 . . . . . 6 (((𝜑𝜒) ∧ ¬ 𝜏) → (¬ 𝜏 → ¬ 𝜒))
118, 10mpd 15 . . . . 5 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜒)
127, 11pm2.65da 816 . . . 4 ((𝜑𝜒) → ¬ ¬ 𝜏)
1312notnotrd 133 . . 3 ((𝜑𝜒) → 𝜏)
1413olcd 873 . 2 ((𝜑𝜒) → (𝜃𝜏))
15 ex-natded5.13.1 . 2 (𝜑 → (𝜓𝜒))
165, 14, 15mpjaodan 958 1 (𝜑 → (𝜃𝜏))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846
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 398  df-or 847
This theorem is referenced by: (None)
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