Theorem 19.38b 1844

Description: Under a nonfreeness hypothesis, the implication 19.38 1842 can be strengthened to an equivalence. See also 19.38a 1843. (Contributed by BJ, 3-Nov-2021.) (Proof shortened by Wolf Lammen, 9-Jul-2022.)

Assertion

Ref	Expression
19.38b	⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))

Proof of Theorem 19.38b

Step	Hyp	Ref	Expression
1		19.38 1842	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
2		exim 1837	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
3		id 22	. . . 4 ⊢ (Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓)
4	3	nfrd 1795	. . 3 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → ∀𝑥𝜓))
5	2, 4	syl9r 78	. 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
6	1, 5	impbid2 225	1 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1783  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-ex 1784  df-nf 1788

This theorem is referenced by:  19.23t  2206