Theorem 19.23h 1802

Description: Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)

Hypothesis

Ref	Expression
19.23h.1	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
19.23h	⊢ (∀x(φ → ψ) ↔ (∃xφ → ψ))

Proof of Theorem 19.23h

Step	Hyp	Ref	Expression
1		19.23h.1	. . 3 ⊢ (ψ → ∀xψ)
2	1	nfi 1551	. 2 ⊢ Ⅎxψ
3	2	19.23 1801	1 ⊢ (∀x(φ → ψ) ↔ (∃xφ → ψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀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  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  exlimihOLD  1805  equsalhw  1838