Theorem nfor 1585

Description: If x is not free in ph and ps, it is not free in ( ph \/ ps ). (Contributed by Jim Kingdon, 11-Mar-2018.)

Hypotheses

Ref	Expression
nfor.1	|- F/ x ph
nfor.2	|- F/ x ps

Assertion

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

Proof of Theorem nfor

Step	Hyp	Ref	Expression
1		nfor.1	. . . 4 |- F/ x ph
2	1	nfri 1530	. . 3 |- ( ph -> A. x ph )
3		nfor.2	. . . 4 |- F/ x ps
4	3	nfri 1530	. . 3 |- ( ps -> A. x ps )
5	2, 4	hbor 1557	. 2 |- ( ( ph \/ ps ) -> A. x ( ph \/ ps ) )
6	5	nfi 1473	1 |- F/ x ( ph \/ ps )

Syntax hints:   \/ wo 709   F/wnf 1471

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-io 710  ax-5 1458  ax-gen 1460  ax-4 1521

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  nfdc  1670  nfun  3315  nfpr  3668  nfso  4333  nffrec  6449  indpi  7402  nfsum1  11499  nfsum  11500  nfcprod1  11697  nfcprod  11698  bj-findis  15476  isomninnlem  15525  trirec0  15539