Theorem 19.38 1871

Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under nonfreeness conditions, see 19.38a 1872 and 19.38b 1873. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2242. (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 1813	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		pm2.21 124	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	2	alimi 1843	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝜓))
4	1, 3	sylbir 238	. 2 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝜓))
5		ala1 1845	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 → 𝜓))
6	4, 5	ja 188	1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  19.38a  1872  19.38b  1873  nfimd  1926  19.21v  1971  19.23v  1974  bj-nfimexal  37583  bj-nfimt  37597  bj-alextruim  37611  bj-wnf1  37696  bj-substax12  37701  bj-nnfim  37729  bj-19.21t  37738  bj-19.23t  37739  bj-19.21t0  37817  pm10.53  45426