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Theorem hbe1 2131
Description: The setvar 𝑥 is not free in 𝑥𝜑. Corresponds to the axiom (5) of modal logic (see also modal5 2144). (Contributed by NM, 24-Jan-1993.)
Assertion
Ref Expression
hbe1 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)

Proof of Theorem hbe1
StepHypRef Expression
1 df-ex 1774 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2 hbn1 2130 . 2 (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
31, 2hbxfrbi 1819 1 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  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-10 2129
This theorem depends on definitions:  df-bi 206  df-ex 1774
This theorem is referenced by:  nfe1  2139  equs5eALT  2356  nfeqf2  2368  equs5e  2449  axie1  2689  bj-wnf2  36087  bj-nnfe1  36129  ac6s6  37534  exlimexi  43799  vk15.4j  43803  vk15.4jVD  44189
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