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Theorem bj-modal5e 32620
Description: Dual statement of hbe1 2020 (which is the real modal-5 2031). See also axc7 2131 and axc7e 2132. (Contributed by BJ, 21-Dec-2020.)
Assertion
Ref Expression
bj-modal5e (∃𝑥𝑥𝜑 → ∀𝑥𝜑)

Proof of Theorem bj-modal5e
StepHypRef Expression
1 hbn1 2019 . . 3 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
2 alnex 1705 . . 3 (∀𝑥 ¬ ∀𝑥𝜑 ↔ ¬ ∃𝑥𝑥𝜑)
31, 2sylib 208 . 2 (¬ ∀𝑥𝜑 → ¬ ∃𝑥𝑥𝜑)
43con4i 113 1 (∃𝑥𝑥𝜑 → ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1480  wex 1703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-10 2018
This theorem depends on definitions:  df-bi 197  df-ex 1704
This theorem is referenced by:  bj-19.41al  32621  bj-sb56  32623
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