Theorem 19.23hOLD 1820

Description: Obsolete proof of 19.23h 1802 as of 1-Jan-2018. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
19.23hOLD.1	⊢ (ψ → ∀xψ)

Assertion

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

Proof of Theorem 19.23hOLD

Step	Hyp	Ref	Expression
1		exim 1575	. . 3 ⊢ (∀x(φ → ψ) → (∃xφ → ∃xψ))
2		19.23hOLD.1	. . . 4 ⊢ (ψ → ∀xψ)
3	2	19.9h 1780	. . 3 ⊢ (∃xψ ↔ ψ)
4	1, 3	syl6ib 217	. 2 ⊢ (∀x(φ → ψ) → (∃xφ → ψ))
5		hbe1 1731	. . . 4 ⊢ (∃xφ → ∀x∃xφ)
6	5, 2	hbim 1817	. . 3 ⊢ ((∃xφ → ψ) → ∀x(∃xφ → ψ))
7		19.8a 1756	. . . 4 ⊢ (φ → ∃xφ)
8	7	imim1i 54	. . 3 ⊢ ((∃xφ → ψ) → (φ → ψ))
9	6, 8	alrimih 1565	. 2 ⊢ ((∃xφ → ψ) → ∀x(φ → ψ))
10	4, 9	impbii 180	1 ⊢ (∀x(φ → ψ) ↔ (∃xφ → ψ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541

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

This theorem is used by: (None)