Theorem nfor 1567

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 1512	. . 3 |- ( ph -> A. x ph )
3		nfor.2	. . . 4 |- F/ x ps
4	3	nfri 1512	. . 3 |- ( ps -> A. x ps )
5	2, 4	hbor 1539	. 2 |- ( ( ph \/ ps ) -> A. x ( ph \/ ps ) )
6	5	nfi 1455	1 |- F/ x ( ph \/ ps )

Syntax hints:   \/ wo 703   F/wnf 1453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-gen 1442  ax-4 1503

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  nfdc  1652  nfun  3283  nfpr  3633  nfso  4287  nffrec  6375  indpi  7304  nfsum1  11319  nfsum  11320  nfcprod1  11517  nfcprod  11518  bj-findis  14014  isomninnlem  14062  trirec0  14076