Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ex-natded5.13 Structured version   Visualization version   GIF version

Theorem ex-natded5.13 27847
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 27848. 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 474 to move it into the ND hypothesis
39 𝜏 → ¬ 𝜒) (𝜑 → (¬ 𝜏 → ¬ 𝜒)) Given \$e. ad2antrr 716 to move it into the ND sub-hypothesis
41 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 479
54 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15 1,3
65 ... (𝜃𝜏) ((𝜑𝜓) → (𝜃𝜏)) I 5 orcd 862 4
76 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 479
88 ... ...| ¬ 𝜏 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜏) (sub) ND hypothesis assumption simpr 479
911 ... ... ¬ 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜒) E 3,8 mpd 15 8,10
107 ... ... 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → 𝜒) IT 7 adantr 474 6
1112 ... ¬ ¬ 𝜏 ((𝜑𝜒) → ¬ ¬ 𝜏) ¬I 8,9,10 pm2.65da 807 7,11
1213 ... 𝜏 ((𝜑𝜒) → 𝜏) ¬E 11 notnotrd 131 12
1314 ... (𝜃𝜏) ((𝜑𝜒) → (𝜃𝜏)) I 12 olcd 863 13
1416 (𝜃𝜏) (𝜑 → (𝜃𝜏)) E 1,6,13 mpjaodan 944 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 474; simpr 479 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 479 . . . 4 ((𝜑𝜓) → 𝜓)
2 ex-natded5.13.2 . . . . 5 (𝜑 → (𝜓𝜃))
32adantr 474 . . . 4 ((𝜑𝜓) → (𝜓𝜃))
41, 3mpd 15 . . 3 ((𝜑𝜓) → 𝜃)
54orcd 862 . 2 ((𝜑𝜓) → (𝜃𝜏))
6 simpr 479 . . . . . 6 ((𝜑𝜒) → 𝜒)
76adantr 474 . . . . 5 (((𝜑𝜒) ∧ ¬ 𝜏) → 𝜒)
8 simpr 479 . . . . . 6 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜏)
9 ex-natded5.13.3 . . . . . . 7 (𝜑 → (¬ 𝜏 → ¬ 𝜒))
109ad2antrr 716 . . . . . 6 (((𝜑𝜒) ∧ ¬ 𝜏) → (¬ 𝜏 → ¬ 𝜒))
118, 10mpd 15 . . . . 5 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜒)
127, 11pm2.65da 807 . . . 4 ((𝜑𝜒) → ¬ ¬ 𝜏)
1312notnotrd 131 . . 3 ((𝜑𝜒) → 𝜏)
1413olcd 863 . 2 ((𝜑𝜒) → (𝜃𝜏))
15 ex-natded5.13.1 . 2 (𝜑 → (𝜓𝜒))
165, 14, 15mpjaodan 944 1 (𝜑 → (𝜃𝜏))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   ∨ wo 836 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 387  df-or 837 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator