Theorem nfbi 1642

Description: If x is not free in ph and ps, then it is not free in ( ph <-> ps ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Hypotheses

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

Assertion

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

Proof of Theorem nfbi

Step	Hyp	Ref	Expression
1		nfbi.1	. . . 4 |- F/ x ph
2	1	a1i 9	. . 3 |- ( T. -> F/ x ph )
3		nfbi.2	. . . 4 |- F/ x ps
4	3	a1i 9	. . 3 |- ( T. -> F/ x ps )
5	2, 4	nfbid 1641	. 2 |- ( T. -> F/ x ( ph <-> ps ) )
6	5	mptru 1411	1 |- F/ x ( ph <-> ps )

This proof depends on syntax axioms:   <-> wb 105   T. wtru 1403   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-5 1500  ax-gen 1502  ax-4 1563  ax-ial 1587  ax-i5r 1588

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514

This theorem is used by:  sb8eu  2099  nfeuv  2104  bm1.1  2223  abbibcom  2352  abbib  2356  nfeq  2400  cleqf  2417  sbhypf  2872  ceqsexg  2954  elabgt  2967  elabgf  2968  copsex2t  4383  copsex2g  4384  opelopabsb  4400  opeliunxp2  4918  ralxpf  4924  rexxpf  4925  cbviota  5340  sb8iota  5343  fmptco  5868  nfiso  6006  uchoice  6365  dfoprab4f  6421  opeliunxp2f  6503  xpf1o  7138  bdsepnfALT  16898