Theorem 19.38a 1869

Description: Under a nonfreeness hypothesis, the implication 19.38 1868 can be strengthened to an equivalence. See also 19.38b 1870. (Contributed by BJ, 3-Nov-2021.) (Proof shortened by Wolf Lammen, 9-Jul-2022.)

Assertion

Ref	Expression
19.38a	⊢ (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))

Proof of Theorem 19.38a

Step	Hyp	Ref	Expression
1		19.38 1868	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
2		id 23	. . . 4 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3	2	nfrd 1820	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
4		alim 1839	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
5	3, 4	syl9 78	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
6	1, 5	impbid2 229	1 ⊢ (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567  ∃wex 1808  Ⅎwnf 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838

This proof depends on definitions:  df-bi 210  df-ex 1809  df-nf 1813

This theorem is used by:  19.21t  2241