Metamath Proof Explorer


Theorem 19.38

Description: Theorem 19.38 of Margaris p. 90. The converse holds under nonfreeness 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 ( 𝜑𝜓 ) )

Proof

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