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  3340  sbcralt  3824  sbcrext  3825  csbiebt  3880  riota5f  7334  axrepndlem1  10486  axrepndlem2  10487  axunnd  10490  axpowndlem2  10492  axpowndlem3  10493  axpowndlem4  10494  axregndlem2  10497  axinfndlem1  10499  axinfnd  10500  axacndlem4  10504  axacndlem5  10505  axacnd  10506  fproddivf  15894  nfan1c  35056  wl-sbcom2d-lem1  37553  wl-mo2df  37564  wl-eudf  37566  wl-mo3t  37570  wl-ax11-lem4  37582  wl-ax11-lem6  37584