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Theorem bj-wnfnf 36741
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 36748, bj-nnfe1 36762 and bj-nnfa1 36761. (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 36720 . 2 (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
2 bj-wnf1 36719 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
3 df-bj-nnf 36726 . 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 1537  wex 1778  Ⅎ'wnnf 36725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-12 2176
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-nf 1783  df-bj-nnf 36726
This theorem is referenced by: (None)
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