Theorem 19.38 1841

Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under nonfreeness conditions, see 19.38a 1842 and 19.38b 1843. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2199. (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 1784	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		pm2.21 123	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	2	alimi 1814	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝜓))
4	1, 3	sylbir 234	. 2 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝜓))
5		ala1 1816	. 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 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  19.38a  1842  19.38b  1843  nfimd  1897  19.21v  1942  19.23v  1945  bj-nfimexal  34807  bj-nfimt  34819  bj-wnf1  34899  bj-substax12  34903  bj-nnfim  34928  bj-19.21t  34951  bj-19.23t  34952  bj-19.21t0  35013  pm10.53  41984