Theorem nfand 1899

Description: If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 ∧ 𝜒). (Contributed by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
nfand.1	⊢ (𝜑 → Ⅎ𝑥𝜓)
nfand.2	⊢ (𝜑 → Ⅎ𝑥𝜒)

Assertion

Ref	Expression
nfand	⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒))

Proof of Theorem nfand

Step	Hyp	Ref	Expression
1		df-an 396	. 2 ⊢ ((𝜓 ∧ 𝜒) ↔ ¬ (𝜓 → ¬ 𝜒))
2		nfand.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3		nfand.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒)
4	3	nfnd 1860	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜒)
5	2, 4	nfimd 1896	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → ¬ 𝜒))
6	5	nfnd 1860	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ (𝜓 → ¬ 𝜒))
7	1, 6	nfxfrd 1856	1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒))

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

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:  nf3and  1900  nfan  1901  nfbid  1904  nfeud2  2590  nfeudw  2591  nfeld  2910  nfrmod  3385  nfreud  3386  nfrmo  3387  nfrab  3427  nfifd  4496  nfdisjw  5064  nfdisj  5065  nfopabd  5153  dfid3  5529  nfriotadw  7332  nfriotad  7335  axrepndlem1  10515  axrepndlem2  10516  axunndlem1  10518  axunnd  10519  axregndlem2  10526  axinfndlem1  10528  axinfnd  10529  axacndlem4  10533  axacndlem5  10534  axacnd  10535  nfchnd  18577  axsepg2  35225  axsepg2ALT  35226  axtcond  36660  bj-gabima  37247  cbvreud  37689  riotasv2d  39403