Theorem 19.21tOLD 1863

Description: Obsolete proof of 19.21t 1795 as of 30-Dec-2017. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
19.21tOLD	⊢ (Ⅎxφ → (∀x(φ → ψ) ↔ (φ → ∀xψ)))

Proof of Theorem 19.21tOLD

Step	Hyp	Ref	Expression
1		id 19	. . . 4 ⊢ (Ⅎxφ → Ⅎxφ)
2	1	nfrd 1763	. . 3 ⊢ (Ⅎxφ → (φ → ∀xφ))
3		alim 1558	. . 3 ⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))
4	2, 3	syl9 66	. 2 ⊢ (Ⅎxφ → (∀x(φ → ψ) → (φ → ∀xψ)))
5		nfa1 1788	. . . . . 6 ⊢ Ⅎx∀xψ
6	5	a1i 10	. . . . 5 ⊢ (Ⅎxφ → Ⅎx∀xψ)
7	1, 6	nfimd 1808	. . . 4 ⊢ (Ⅎxφ → Ⅎx(φ → ∀xψ))
8	7	nfrd 1763	. . 3 ⊢ (Ⅎxφ → ((φ → ∀xψ) → ∀x(φ → ∀xψ)))
9		sp 1747	. . . . 5 ⊢ (∀xψ → ψ)
10	9	imim2i 13	. . . 4 ⊢ ((φ → ∀xψ) → (φ → ψ))
11	10	alimi 1559	. . 3 ⊢ (∀x(φ → ∀xψ) → ∀x(φ → ψ))
12	8, 11	syl6 29	. 2 ⊢ (Ⅎxφ → ((φ → ∀xψ) → ∀x(φ → ψ)))
13	4, 12	impbid 183	1 ⊢ (Ⅎxφ → (∀x(φ → ψ) ↔ (φ → ∀xψ)))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544

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: (None)