Theorem 19.38 1838

Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under nonfreeness conditions, see 19.38a 1839 and 19.38b 1840. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2205. (Revised by Wolf Lammen, 2-Jan-2018.)

Assertion

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

Proof of Theorem 19.38

Step	Hyp	Ref	Expression
1		alnex 1780	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		pm2.21 123	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	2	alimi 1810	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝜓))
4	1, 3	sylbir 235	. 2 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝜓))
5		ala1 1812	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 → 𝜓))
6	4, 5	ja 186	1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808

This theorem depends on definitions:  df-bi 207  df-ex 1779

This theorem is referenced by:  19.38a  1839  19.38b  1840  nfimd  1893  19.21v  1938  19.23v  1941  bj-nfimexal  36628  bj-nfimt  36640  bj-wnf1  36719  bj-substax12  36723  bj-nnfim  36748  bj-19.21t  36771  bj-19.23t  36772  bj-19.21t0  36832  pm10.53  44390