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Theorem hbe1 2143
Description: The setvar 𝑥 is not free in 𝑥𝜑. Corresponds to the axiom (5) of modal logic (see also modal5 2156). (Contributed by NM, 24-Jan-1993.)
Assertion
Ref Expression
hbe1 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)

Proof of Theorem hbe1
StepHypRef Expression
1 df-ex 1787 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2 hbn1 2142 . 2 (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
31, 2hbxfrbi 1831 1 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1540  wex 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-10 2141
This theorem depends on definitions:  df-bi 206  df-ex 1787
This theorem is referenced by:  nfe1  2151  equs5eALT  2367  nfeqf2  2379  equs5e  2460  axie1  2705  bj-wnf2  34894  bj-nnfe1  34936  ac6s6  36324  exlimexi  42112  vk15.4j  42116  vk15.4jVD  42502
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