Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-wnfnf Structured version   Visualization version   GIF version

Theorem bj-wnfnf 36722
Description: When 𝜑 is substituted for 𝜓, this statement expresses nonfreeness in the weak form of nonfreeness (∃ → ∀). Note that this could also be proved from bj-nnfim 36729, bj-nnfe1 36743 and bj-nnfa1 36742. (Contributed by BJ, 9-Dec-2023.)
Assertion
Ref Expression
bj-wnfnf Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)

Proof of Theorem bj-wnfnf
StepHypRef Expression
1 bj-wnf2 36701 . 2 (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
2 bj-wnf1 36700 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
3 df-bj-nnf 36707 . 2 (Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓) ↔ ((∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)) ∧ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))))
41, 2, 3mpbir2an 711 1 Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535  wex 1776  Ⅎ'wnnf 36706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-bj-nnf 36707
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator