Theorem hbe1 2167

Description: The setvar 𝑥 is not free in ∃𝑥𝜑. Corresponds to the axiom (5) of modal logic (see also modal5 2179). (Contributed by NM, 24-Jan-1993.)

Assertion

Ref	Expression
hbe1	⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)

Proof of Theorem hbe1

Step	Hyp	Ref	Expression
1		df-ex 1790	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2		hbn1 2166	. 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
3	1, 2	hbxfrbi 1835	1 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1548  ∃wex 1789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-10 2165

This theorem depends on definitions:  df-bi 209  df-ex 1790

This theorem is referenced by:  nfe1  2174  nfexhe  2200  equs5eALT  2388  nfeqf2  2398  equs5e  2479  axie1  2718  bj-wnf2  37133  bj-nnfe1  37198  ac6s6  38609  nfale2  42771  exlimexi  45038  vk15.4j  45042  vk15.4jVD  45427