Theorem hbe1 1731

Description: x is not free in ∃xφ. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
hbe1	⊢ (∃xφ → ∀x∃xφ)

Proof of Theorem hbe1

Step	Hyp	Ref	Expression
1		df-ex 1542	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
2		hbn1 1730	. 2 ⊢ (¬ ∀x ¬ φ → ∀x ¬ ∀x ¬ φ)
3	1, 2	hbxfrbi 1568	1 ⊢ (∃xφ → ∀x∃xφ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-6 1729

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  nfe1  1732  hba1  1786  19.23hOLD  1820  ax12olem5  1931  ax10lem2  1937  axie1  2328  hboprab1  5560  hboprab2  5561  hboprab3  5562