Theorem 19.38 1934

Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under non-freeness conditions, see 19.38a 1935 and 19.38b 1937. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2240. (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 1877	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		pm2.21 121	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	2	alimi 1907	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝜓))
4	1, 3	sylbir 227	. 2 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝜓))
5		ala1 1909	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 → 𝜓))
6	4, 5	ja 175	1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1651  ∃wex 1875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905

This theorem depends on definitions:  df-bi 199  df-ex 1876

This theorem is referenced by:  19.38a  1935  19.38aOLD  1936  19.38b  1937  19.38bOLD  1938  nfimd  1993  19.21v  2035  19.23v  2038  19.23vOLD  2086  bj-nfimt  33122  bj-19.21t  33312  pm10.53  39343