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Theorem bj-wnfanf 36662
Description: When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the universal form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
Assertion
Ref Expression
bj-wnfanf ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))

Proof of Theorem bj-wnfanf
StepHypRef Expression
1 bj-wnf1 36660 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
2 bj-19.23bit 36634 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜓))
31, 2sylg 1818 1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1533  wex 1774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-10 2137  ax-12 2173
This theorem depends on definitions:  df-bi 207  df-or 847  df-ex 1775  df-nf 1779
This theorem is referenced by: (None)
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