Theorem hbe1 2187

Description: The setvar 𝑥 is not free in ∃𝑥𝜑. Corresponds to the axiom (5) of modal logic (see also modal5 2199). (Contributed by NM, 24-Jan-1993.)

Assertion

Ref	Expression
hbe1	⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)

Proof of Theorem hbe1

Step	Hyp	Ref	Expression
1		df-ex 1876	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2		hbn1 2186	. 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
3	1, 2	hbxfrbi 1920	1 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1651  ∃wex 1875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-10 2185

This theorem depends on definitions:  df-bi 199  df-ex 1876

This theorem is referenced by:  nfe1  2194  equs5eALT  2375  nfeqf2  2382  nfeqf2OLD  2383  equs5e  2465  axie1  2771  ac6s6  34466  exlimexi  39510  vk15.4j  39514  vk15.4jVD  39910