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Theorem bj-wnf2 34072
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-wnf2 (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem bj-wnf2
StepHypRef Expression
1 hbe1 2148 . 2 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
2 bj-eximcom 33996 . 2 (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∀𝑥𝑥𝜑 → ∃𝑥𝑥𝜓))
3 hbe1a 2149 . 2 (∃𝑥𝑥𝜓 → ∀𝑥𝜓)
41, 2, 3syl56 36 1 (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-10 2146
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by:  bj-wnfnf  34090
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