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Theorem bj-nnflemee 34953
Description: One of four lemmas for nonfreeness: antecedent and consequent both expressed using existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nnflemee (∀𝑥(∃𝑦𝜑𝜑) → (∃𝑦𝑥𝜑 → ∃𝑥𝜑))

Proof of Theorem bj-nnflemee
StepHypRef Expression
1 excom 2162 . 2 (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
2 exim 1836 . 2 (∀𝑥(∃𝑦𝜑𝜑) → (∃𝑥𝑦𝜑 → ∃𝑥𝜑))
31, 2syl5bi 241 1 (∀𝑥(∃𝑦𝜑𝜑) → (∃𝑦𝑥𝜑 → ∃𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1537  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-11 2154
This theorem depends on definitions:  df-bi 206  df-ex 1783
This theorem is referenced by:  bj-nnfext  34957
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