Theorem hbe1 2179

Description: The setvar 𝑥 is not free in ∃𝑥𝜑. Corresponds to the axiom (5) of modal logic (see also modal5 2191). (Contributed by NM, 24-Jan-1993.)

Assertion

Ref	Expression
hbe1	⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)

Proof of Theorem hbe1

Step	Hyp	Ref	Expression
1		df-ex 1812	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2		hbn1 2178	. 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
3	1, 2	hbxfrbi 1857	1 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-10 2177

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  nfe1  2186  nfexhe  2210  equs5eALT  2396  nfeqf2  2406  equs5e  2487  axie1  2726  bj-wnf2  37715  bj-nnfe1  37780  ac6s6  39198  nfale2  43362  exlimexi  45601  vk15.4j  45605  vk15.4jVD  45990