Theorem nfand 1898

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 1859	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜒)
5	2, 4	nfimd 1895	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → ¬ 𝜒))
6	5	nfnd 1859	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ (𝜓 → ¬ 𝜒))
7	1, 6	nfxfrd 1855	1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785

This theorem is referenced by:  nf3and  1899  nfan  1900  nfbid  1903  nfeud2  2590  nfeudw  2591  nfeld  2910  nfrmod  3395  nfreud  3396  nfrmo  3397  nfrab  3438  nfifd  4509  nfdisjw  5077  nfdisj  5078  nfopabd  5166  dfid3  5522  nfriotadw  7323  nfriotad  7326  axrepndlem1  10503  axrepndlem2  10504  axunndlem1  10506  axunnd  10507  axregndlem2  10514  axinfndlem1  10516  axinfnd  10517  axacndlem4  10521  axacndlem5  10522  axacnd  10523  nfchnd  18534  axsepg2  35238  axsepg2ALT  35239  bj-gabima  37141  cbvreud  37574  riotasv2d  39213