Theorem nfan1 2193

Description: A closed form of nfan 1902. (Contributed by Mario Carneiro, 3-Oct-2016.) df-nf 1787 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 397	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
2		nfim1.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3		nfim1.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	3	nfnd 1861	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
5	2, 4	nfim1 2192	. . 3 ⊢ Ⅎ𝑥(𝜑 → ¬ 𝜓)
6	5	nfn 1860	. 2 ⊢ Ⅎ𝑥 ¬ (𝜑 → ¬ 𝜓)
7	1, 6	nfxfr 1855	1 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787

This theorem is referenced by:  sb4b  2475  sb4bOLD  2476  ralcom2  3290  sbcralt  3805  sbcrext  3806  csbiebt  3862  riota5f  7261  axrepndlem1  10348  axrepndlem2  10349  axunnd  10352  axpowndlem2  10354  axpowndlem3  10355  axpowndlem4  10356  axregndlem2  10359  axinfndlem1  10361  axinfnd  10362  axacndlem4  10366  axacndlem5  10367  axacnd  10368  fproddivf  15697  wl-sbcom2d-lem1  35714  wl-mo2df  35725  wl-eudf  35727  wl-mo3t  35731  wl-ax11-lem4  35739  wl-ax11-lem6  35741