Theorem nfan1c 35461

Description: Variant of nfan 1929 and commuted form of nfan1 2236. (Contributed by BTernaryTau, 31-Jul-2025.)

Hypotheses

Ref	Expression
nfan1c.1	⊢ Ⅎ𝑥𝜑
nfan1c.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfan1c	⊢ Ⅎ𝑥(𝜓 ∧ 𝜑)

Proof of Theorem nfan1c

Step	Hyp	Ref	Expression
1		nfan1c.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		nfan1c.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3	1, 2	nfan1 2236	. 2 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)
4		ancom 465	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
5	4	nfbii 1882	. 2 ⊢ (Ⅎ𝑥(𝜑 ∧ 𝜓) ↔ Ⅎ𝑥(𝜓 ∧ 𝜑))
6	3, 5	mpbi 233	1 ⊢ Ⅎ𝑥(𝜓 ∧ 𝜑)

Syntax hints:   → wi 4   ∧ wa 400  Ⅎwnf 1813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1810  df-nf 1814

This theorem is referenced by:  dvelimalcased  35463  dvelimexcased  35465