Theorem nfbid 1477

Description: If in a context x is not free in ps and ch, it is not free in (ps <-> ch). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)

Hypotheses

Ref	Expression
nfbid.1	|- (ph -> F/xps)
nfbid.2	|- (ph -> F/xch)

Assertion

Ref	Expression
nfbid	|- (ph -> F/x(ps <-> ch))

Proof of Theorem nfbid

Step	Hyp	Ref	Expression
1		dfbi2 368	. 2 |- ((ps <-> ch) <-> ((ps -> ch) /\ (ch -> ps)))
2		nfbid.1	. . . 4 |- (ph -> F/xps)
3		nfbid.2	. . . 4 |- (ph -> F/xch)
4	2, 3	nfimd 1474	. . 3 |- (ph -> F/x(ps -> ch))
5	3, 2	nfimd 1474	. . 3 |- (ph -> F/x(ch -> ps))
6	4, 5	nfand 1457	. 2 |- (ph -> F/x((ps -> ch) /\ (ch -> ps)))
7	1, 6	nfxfrd 1361	1 |- (ph -> F/x(ps <-> ch))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   F/wnf 1346

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 1333  ax-gen 1335  ax-4 1397  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347

This theorem is referenced by:  nfbi  1478  nfeudv  1912  nfeqd  2189  nfiotadxy  4813  iota2df  4834  bdsepnft  9321  strcollnft  9414