Theorem nf3an

Description: If x is not free in ph , ps , and ch , then it is not free in ( ph /\ ps /\ ch ) . (Contributed by Mario Carneiro, 11-Aug-2016)

	Ref	Expression
Hypotheses	nfan.1	$$⊢Ⅎx\phi$$
	nfan.2	$$⊢Ⅎx\psi$$
	nfan.3	$$⊢Ⅎx\chi$$
Assertion	nf3an	$$⊢Ⅎx\left(\phi \wedge \psi \wedge \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfan.1	$$⊢Ⅎx\phi$$
2		nfan.2	$$⊢Ⅎx\psi$$
3		nfan.3	$$⊢Ⅎx\chi$$
4		df-3an	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\wedge \chi \right)$$
5	1 2	nfan	$$⊢Ⅎx\left(\phi \wedge \psi \right)$$
6	5 3	nfan	$$⊢Ⅎx\left(\left(\phi \wedge \psi \right)\wedge \chi \right)$$
7	4 6	nfxfr	$$⊢Ⅎx\left(\phi \wedge \psi \wedge \chi \right)$$