Theorem nfan1 1562

Description: A closed form of nfan 1563. (Contributed by Mario Carneiro, 3-Oct-2016.)

Hypotheses

Ref	Expression
nfan1.1	⊢ Ⅎ𝑥𝜑
nfan1.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfan1	⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)

Proof of Theorem nfan1

Step	Hyp	Ref	Expression
1		nfan1.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
2	1	nfrd 1518	. . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
3	2	imdistani 445	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ ∀𝑥𝜓))
4		nfan1.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
5	4	nfri 1517	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
6	5	19.28h 1560	. . 3 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
7	3, 6	sylibr 134	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))
8	7	nfi 1460	1 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1351  Ⅎwnf 1458

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 1445  ax-gen 1447  ax-4 1508

This theorem depends on definitions:  df-bi 117  df-nf 1459

This theorem is referenced by:  nfan  1563  sbcralt  3037  sbcrext  3038  csbiebt  3094  riota5f  5845  fproddivapf  11607