Theorem nf3an 1589

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 983	. 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 1588	. . 3 |- F/ x ( ph /\ ps )
5		nfan.3	. . 3 |- F/ x ch
6	4, 5	nfan 1588	. 2 |- F/ x ( ( ph /\ ps ) /\ ch )
7	1, 6	nfxfr 1497	1 |- F/ x ( ph /\ ps /\ ch )

Syntax hints:   /\ wa 104   /\ w3a 981   F/wnf 1483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-gen 1472  ax-4 1533

This theorem depends on definitions:  df-bi 117  df-3an 983  df-nf 1484

This theorem is referenced by:  vtocl3gaf  2842  mob  2955  nfop  3835  mkvprop  7260  seq3f1olemstep  10659  seq3f1olemp  10660  nfsum1  11667  nfsum  11668  dfgrp3mlem  13430