Theorem 19.23ht 1386

Description: Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 1-Feb-2015.)

Assertion

Ref	Expression
19.23ht	⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))

Proof of Theorem 19.23ht

Step	Hyp	Ref	Expression
1		ax-ie2 1383	1 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  ∃wex 1381

This theorem was proved from axioms:  ax-ie2 1383

This theorem is referenced by:  19.23h  1387  exlimd2  1486  19.9ht  1532