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Theorem hbe1 2019
Description: The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by NM, 24-Jan-1993.)
Assertion
Ref Expression
hbe1 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)

Proof of Theorem hbe1
StepHypRef Expression
1 df-ex 1703 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2 hbn1 2018 . 2 (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
31, 2hbxfrbi 1750 1 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1479  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-10 2017
This theorem depends on definitions:  df-bi 197  df-ex 1703
This theorem is referenced by:  nfe1  2025  nfe1OLD  2031  equs5eALT  2176  equs5e  2347  axie1  2594  wl-dveeq12  33282  ac6s6  33951  exlimexi  38550  vk15.4j  38554  vk15.4jVD  38970
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