Theorem hbe1 2154

Description: The setvar 𝑥 is not free in ∃𝑥𝜑. Corresponds to the axiom (5) of modal logic (see also modal5 2166). (Contributed by NM, 24-Jan-1993.)

Assertion

Ref	Expression
hbe1	⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)

Proof of Theorem hbe1

Step	Hyp	Ref	Expression
1		df-ex 1787	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2		hbn1 2153	. 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
3	1, 2	hbxfrbi 1832	1 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-10 2152

This theorem depends on definitions:  df-bi 208  df-ex 1787

This theorem is referenced by:  nfe1  2161  nfexhe  2187  equs5eALT  2375  nfeqf2  2385  equs5e  2466  axie1  2706  bj-wnf2  37070  bj-nnfe1  37135  ac6s6  38546  nfale2  42708  exlimexi  44975  vk15.4j  44979  vk15.4jVD  45364