Theorem 19.9t 719

Description: A closed version of one direction of 19.9r 718.

Assertion

Ref	Expression
19.9t	⊢ (∀x(φ → ∀xφ) → (∃xφ → φ))

Proof of Theorem 19.9t

Step	Hyp	Ref	Expression
1		hbnt 710	. . 3 ⊢ (∀x(φ → ∀xφ) → (¬ φ → ∀x ¬ φ))
2	1	con1d 85	. 2 ⊢ (∀x(φ → ∀xφ) → (¬ ∀x ¬ φ → φ))
3		df-ex 679	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
4	2, 3	syl5ib 181	1 ⊢ (∀x(φ → ∀xφ) → (∃xφ → φ))

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672  ∃wex 678

This theorem is referenced by:  19.9d 720  exists2 1073

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-ex 679

metamath.org