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Theorem bj-wnfanf 34669
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 34667 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
2 bj-19.23bit 34641 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜓))
31, 2sylg 1830 1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1541  wex 1787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2143  ax-12 2177
This theorem depends on definitions:  df-bi 210  df-or 848  df-ex 1788  df-nf 1792
This theorem is referenced by: (None)
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