Theorem nfor 1627

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 1572	. . 3 |- ( ph -> A. x ph )
3		nfor.2	. . . 4 |- F/ x ps
4	3	nfri 1572	. . 3 |- ( ps -> A. x ps )
5	2, 4	hbor 1599	. 2 |- ( ( ph \/ ps ) -> A. x ( ph \/ ps ) )
6	5	nfi 1515	1 |- F/ x ( ph \/ ps )

This proof depends on syntax axioms:   \/ wo 720   F/wnf 1513

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-gen 1502  ax-4 1563

This proof depends on definitions:  df-bi 117  df-nf 1514

This theorem is used by:  nfdc  1711  nfun  3385  nfpr  3759  rabsnifsb  3777  nfso  4447  nffrec  6667  indpi  7709  nfsum1  12122  nfsum  12123  nfcprod1  12321  nfcprod  12322  bj-findis  17005  isomninnlem  17079  trirec0  17093