Theorem 19.38

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

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

Proof

Step	Hyp	Ref	Expression
1		alnex	⊢ ( ∀ 𝑥 ¬ 𝜑 ↔ ¬ ∃ 𝑥 𝜑 )
2		pm2.21	⊢ ( ¬ 𝜑 → ( 𝜑 → 𝜓 ) )
3	2	alimi	⊢ ( ∀ 𝑥 ¬ 𝜑 → ∀ 𝑥 ( 𝜑 → 𝜓 ) )
4	1 3	sylbir	⊢ ( ¬ ∃ 𝑥 𝜑 → ∀ 𝑥 ( 𝜑 → 𝜓 ) )
5		ala1	⊢ ( ∀ 𝑥 𝜓 → ∀ 𝑥 ( 𝜑 → 𝜓 ) )
6	4 5	ja	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) )