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Theorem bj-wnfnf 35233
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 35240, bj-nnfe1 35254 and bj-nnfa1 35253. (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 35212 . 2 (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
2 bj-wnf1 35211 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
3 df-bj-nnf 35218 . 2 (Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓) ↔ ((∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)) ∧ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))))
41, 2, 3mpbir2an 710 1 Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wex 1782  Ⅎ'wnnf 35217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-bj-nnf 35218
This theorem is referenced by: (None)
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