Theorem nfim1OLD 1812

Description: A closed form of nfim 1813. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfim1.1	⊢ Ⅎxφ
nfim1.2	⊢ (φ → Ⅎxψ)

Assertion

Ref	Expression
nfim1OLD	⊢ Ⅎx(φ → ψ)

Proof of Theorem nfim1OLD

Step	Hyp	Ref	Expression
1		nfim1.2	. . . . 5 ⊢ (φ → Ⅎxψ)
2	1	nfrd 1763	. . . 4 ⊢ (φ → (ψ → ∀xψ))
3	2	a2i 12	. . 3 ⊢ ((φ → ψ) → (φ → ∀xψ))
4		nfim1.1	. . . 4 ⊢ Ⅎxφ
5	4	19.21 1796	. . 3 ⊢ (∀x(φ → ψ) ↔ (φ → ∀xψ))
6	3, 5	sylibr 203	. 2 ⊢ ((φ → ψ) → ∀x(φ → ψ))
7	6	nfi 1551	1 ⊢ Ⅎx(φ → ψ)

This proof depends on syntax axioms:   → wi 4  ∀wal 1540  Ⅎwnf 1544

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)