Theorem nfrimi 1539

Description: Moving an antecedent outside Ⅎ. (Contributed by Jim Kingdon, 23-Mar-2018.)

Hypotheses

Ref	Expression
nfrimi.1	⊢ Ⅎ𝑥𝜑
nfrimi.2	⊢ Ⅎ𝑥(𝜑 → 𝜓)

Assertion

Ref	Expression
nfrimi	⊢ (𝜑 → Ⅎ𝑥𝜓)

Proof of Theorem nfrimi

Step	Hyp	Ref	Expression
1		nfrimi.1	. 2 ⊢ Ⅎ𝑥𝜑
2		nfrimi.2	. . . . 5 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
3	2	nfri 1533	. . . 4 ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))
4	1	nfri 1533	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
5		ax-5 1461	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
6	3, 4, 5	syl2im 38	. . 3 ⊢ ((𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
7	6	pm2.86i 99	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
8	1, 7	nfd 1537	1 ⊢ (𝜑 → Ⅎ𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1362  Ⅎwnf 1474

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 1461  ax-gen 1463  ax-4 1524

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  hbsbd  2001