Theorem hbe1 2180

Description: The setvar 𝑥 is not free in ∃𝑥𝜑. Corresponds to the axiom (5) of modal logic (see also modal5 2192). (Contributed by NM, 24-Jan-1993.)

Assertion

Ref	Expression
hbe1	⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)

Proof of Theorem hbe1

Step	Hyp	Ref	Expression
1		df-ex 1803	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2		hbn1 2179	. 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
3	1, 2	hbxfrbi 1848	1 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1561  ∃wex 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-10 2178

This theorem depends on definitions:  df-bi 210  df-ex 1803

This theorem is referenced by:  nfe1  2187  nfexhe  2213  equs5eALT  2401  nfeqf2  2411  equs5e  2492  axie1  2731  bj-wnf2  37202  bj-nnfe1  37267  ac6s6  38678  nfale2  42840  exlimexi  45092  vk15.4j  45096  vk15.4jVD  45481