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Theorem bj-nnflemea 36151
Description: One of four lemmas for nonfreeness: antecedent expressed with existential quantifier and consequent expressed with universal quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nnflemea (∀𝑥(∃𝑦𝜑𝜑) → (∃𝑦𝑥𝜑 → ∀𝑥𝜑))

Proof of Theorem bj-nnflemea
StepHypRef Expression
1 bj-19.12 36147 . 2 (∃𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
2 alim 1804 . 2 (∀𝑥(∃𝑦𝜑𝜑) → (∀𝑥𝑦𝜑 → ∀𝑥𝜑))
31, 2syl5 34 1 (∀𝑥(∃𝑦𝜑𝜑) → (∃𝑦𝑥𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-ex 1774
This theorem is referenced by:  bj-nnfalt  36152
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