Theorem nfan1c

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

	Ref	Expression
Hypotheses	nfan1c.1	$$⊢Ⅎx\phi$$
	nfan1c.2	$$⊢\phi \to Ⅎx\psi$$
Assertion	nfan1c	$$⊢Ⅎx\left(\psi \wedge \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfan1c.1	$$⊢Ⅎx\phi$$
2		nfan1c.2	$$⊢\phi \to Ⅎx\psi$$
3	1 2	nfan1	$$⊢Ⅎx\left(\phi \wedge \psi \right)$$
4		ancom	$$⊢\left(\phi \wedge \psi \right)\leftrightarrow \left(\psi \wedge \phi \right)$$
5	4	nfbii	$$⊢Ⅎx\left(\phi \wedge \psi \right)\leftrightarrow Ⅎx\left(\psi \wedge \phi \right)$$
6	3 5	mpbi	$$⊢Ⅎx\left(\psi \wedge \phi \right)$$