Theorem hbe1 2021

Description: The setvar 𝑥 is not free in ∃𝑥𝜑. (Contributed by NM, 24-Jan-1993.)

Assertion

Ref	Expression
hbe1	⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)

Proof of Theorem hbe1

Step	Hyp	Ref	Expression
1		df-ex 1705	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2		hbn1 2020	. 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
3	1, 2	hbxfrbi 1752	1 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-10 2019

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  nfe1  2027  nfe1OLD  2033  equs5eALT  2178  equs5e  2349  axie1  2596  wl-dveeq12  33311  ac6s6  33980  exlimexi  38730  vk15.4j  38734  vk15.4jVD  39150