Theorem nfan1 2201

Description: A closed form of nfan 1899. (Contributed by Mario Carneiro, 3-Oct-2016.) df-nf 1784 changed. (Revised by Wolf Lammen, 18-Sep-2021.) (Proof shortened by Wolf Lammen, 7-Jul-2022.)

Hypotheses

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

Assertion

Ref	Expression
nfan1	⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)

Proof of Theorem nfan1

Step	Hyp	Ref	Expression
1		df-an 396	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
2		nfim1.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3		nfim1.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	3	nfnd 1858	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
5	2, 4	nfim1 2200	. . 3 ⊢ Ⅎ𝑥(𝜑 → ¬ 𝜓)
6	5	nfn 1857	. 2 ⊢ Ⅎ𝑥 ¬ (𝜑 → ¬ 𝜓)
7	1, 6	nfxfr 1853	1 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-nf 1784

This theorem is referenced by:  sb4b  2473  ralcom2  3351  sbcralt  3835  sbcrext  3836  csbiebt  3891  riota5f  7372  axrepndlem1  10545  axrepndlem2  10546  axunnd  10549  axpowndlem2  10551  axpowndlem3  10552  axpowndlem4  10553  axregndlem2  10556  axinfndlem1  10558  axinfnd  10559  axacndlem4  10563  axacndlem5  10564  axacnd  10565  fproddivf  15953  nfan1c  35063  wl-sbcom2d-lem1  37547  wl-mo2df  37558  wl-eudf  37560  wl-mo3t  37564  wl-ax11-lem4  37576  wl-ax11-lem6  37578