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Theorem bj-wnf1 34526
Description: When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.)
Assertion
Ref Expression
bj-wnf1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem bj-wnf1
StepHypRef Expression
1 bj-modal4e 34524 . . 3 (∃𝑥𝑥𝜑 → ∃𝑥𝜑)
2 hba1 2296 . . 3 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
31, 2imim12i 62 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝑥𝜑 → ∀𝑥𝑥𝜓))
4 19.38 1845 . 2 ((∃𝑥𝑥𝜑 → ∀𝑥𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
53, 4syl 17 1 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wex 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-10 2144  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-or 847  df-ex 1787  df-nf 1791
This theorem is referenced by:  bj-wnfanf  34528  bj-wnfenf  34529  bj-wnfnf  34548
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