Theorem nfim1 1624

Description: A closed form of nfim 1625. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Hypotheses

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

Assertion

Ref	Expression
nfim1	⊢ Ⅎ𝑥(𝜑 → 𝜓)

Proof of Theorem nfim1

Step	Hyp	Ref	Expression
1		nfim1.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	nfri 1572	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3		nfim1.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	3	nfrd 1573	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
5	2, 4	hbim1 1623	. 2 ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))
6	5	nfi 1515	1 ⊢ Ⅎ𝑥(𝜑 → 𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514

This theorem is referenced by:  nfim  1625  cbv1  1798  cbv1v  1800  hbsbd  2042  nfabdw  2411