Users' Mathboxes Mathbox for Adrian Ducourtial < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  antnest Structured version   Visualization version   GIF version

Theorem antnest 35673
Description: Suppose 𝜑, 𝜓 are distinct atomic propositional formulas, and let Γ be the smallest class of formulas for which ⊤ ∈ Γ and (𝜒𝜑), (𝜒𝜓) ∈ Γ for 𝜒 ∈ Γ. The present theorem is then an element of Γ, and the implications occurring in the theorem are in one-to-one correspondence with the formulas in Γ up to logical equivalence. In particular, the theorem itself is equivalent to ⊤ ∈ Γ. (Contributed by Adrian Ducourtial, 2-Oct-2025.)
Assertion
Ref Expression
antnest ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)

Proof of Theorem antnest
StepHypRef Expression
1 simplim 167 . . . . 5 (¬ ((⊤ → 𝜑) → 𝜓) → (⊤ → 𝜑))
2 conax1 170 . . . . . 6 (¬ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓) → ¬ 𝜓)
3 simplim 167 . . . . . . . 8 (¬ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓) → (((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓))
42, 3mtod 198 . . . . . . 7 (¬ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓) → ¬ ((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑))
5 simplim 167 . . . . . . 7 (¬ ((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → (((⊤ → 𝜑) → 𝜓) → 𝜓))
64, 5syl 17 . . . . . 6 (¬ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓) → (((⊤ → 𝜑) → 𝜓) → 𝜓))
72, 6mtod 198 . . . . 5 (¬ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓) → ¬ ((⊤ → 𝜑) → 𝜓))
81, 7syl11 33 . . . 4 (⊤ → (¬ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓) → 𝜑))
98mptru 1543 . . 3 (¬ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓) → 𝜑)
10 conax1 170 . . . 4 (¬ ((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → ¬ 𝜑)
114, 10syl 17 . . 3 (¬ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓) → ¬ 𝜑)
129, 11pm2.65i 194 . 2 ¬ ¬ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)
1312notnotri 131 1 ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wtru 1537
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 207  df-tru 1539
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator