Theorem 19.38 1794

Description: Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (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 1543	. . 3 ⊢ (∀x ¬ φ ↔ ¬ ∃xφ)
2		pm2.21 100	. . . 4 ⊢ (¬ φ → (φ → ψ))
3	2	alimi 1559	. . 3 ⊢ (∀x ¬ φ → ∀x(φ → ψ))
4	1, 3	sylbir 204	. 2 ⊢ (¬ ∃xφ → ∀x(φ → ψ))
5		ax-1 6	. . 3 ⊢ (ψ → (φ → ψ))
6	5	alimi 1559	. 2 ⊢ (∀xψ → ∀x(φ → ψ))
7	4, 6	ja 153	1 ⊢ ((∃xφ → ∀xψ) → ∀x(φ → ψ))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  19.21t  1795  19.23t  1800