Theorem nfan1 2208

Description: A closed form of nfan 1901. (Contributed by Mario Carneiro, 3-Oct-2016.) df-nf 1786 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 1860	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
5	2, 4	nfim1 2207	. . 3 ⊢ Ⅎ𝑥(𝜑 → ¬ 𝜓)
6	5	nfn 1859	. 2 ⊢ Ⅎ𝑥 ¬ (𝜑 → ¬ 𝜓)
7	1, 6	nfxfr 1855	1 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786

This theorem is referenced by:  sb4b  2480  ralcom2  3349  sbcralt  3824  sbcrext  3825  csbiebt  3880  riota5f  7355  axrepndlem1  10517  axrepndlem2  10518  axunnd  10521  axpowndlem2  10523  axpowndlem3  10524  axpowndlem4  10525  axregndlem2  10528  axinfndlem1  10530  axinfnd  10531  axacndlem4  10535  axacndlem5  10536  axacnd  10537  fproddivf  15924  nfan1c  35255  mh-setindnd  36695  wl-sbcom2d-lem1  37843  wl-mo2df  37854  wl-eudf  37856  wl-mo3t  37860