Theorem nfan1c 35080

Description: Variant of nfan 1900 and commuted form of nfan1 2203. (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 2203	. 2 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)
4		ancom 460	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
5	4	nfbii 1853	. 2 ⊢ (Ⅎ𝑥(𝜑 ∧ 𝜓) ↔ Ⅎ𝑥(𝜓 ∧ 𝜑))
6	3, 5	mpbi 230	1 ⊢ Ⅎ𝑥(𝜓 ∧ 𝜑)

Syntax hints:   → 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  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2180

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:  dvelimalcased  35082  dvelimexcased  35084