Theorem 19.38 1862

Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under nonfreeness conditions, see 19.38a 1863 and 19.38b 1864. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2244. (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 1804	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		pm2.21 124	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	2	alimi 1834	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝜓))
4	1, 3	sylbir 238	. 2 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝜓))
5		ala1 1836	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 → 𝜓))
6	4, 5	ja 188	1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1561  ∃wex 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832

This theorem depends on definitions:  df-bi 210  df-ex 1803

This theorem is referenced by:  19.38a  1863  19.38b  1864  nfimd  1917  19.21v  1962  19.23v  1965  bj-nfimexal  37093  bj-nfimt  37107  bj-alextruim  37121  bj-wnf1  37206  bj-substax12  37211  bj-nnfim  37239  bj-19.21t  37248  bj-19.23t  37249  bj-19.21t0  37327  pm10.53  44940