Theorem nf3an 1458

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

Hypotheses

Ref	Expression
nfan.1	|- F/ x ph
nfan.2	|- F/ x ps
nfan.3	|- F/ x ch

Assertion

Ref	Expression
nf3an	|- F/ x ( ph /\ ps /\ ch )

Proof of Theorem nf3an

Step	Hyp	Ref	Expression
1		df-3an 887	. 2 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
2		nfan.1	. . . 4 |- F/ x ph
3		nfan.2	. . . 4 |- F/ x ps
4	2, 3	nfan 1457	. . 3 |- F/ x ( ph /\ ps )
5		nfan.3	. . 3 |- F/ x ch
6	4, 5	nfan 1457	. 2 |- F/ x ( ( ph /\ ps ) /\ ch )
7	1, 6	nfxfr 1363	1 |- F/ x ( ph /\ ps /\ ch )

Syntax hints:   /\ wa 97   /\ w3a 885   F/wnf 1349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400

This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350

This theorem is referenced by:  vtocl3gaf  2622  mob  2723  nfop  3565