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Theorem nfbi 1856
Description: If  x is not free in  ph and  ps, 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
nf.1  |-  F/ x ph
nf.2  |-  F/ x ps
Assertion
Ref Expression
nfbi  |-  F/ x
( ph  <->  ps )

Proof of Theorem nfbi
StepHypRef Expression
1 nf.1 . . . 4  |-  F/ x ph
21a1i 11 . . 3  |-  (  T. 
->  F/ x ph )
3 nf.2 . . . 4  |-  F/ x ps
43a1i 11 . . 3  |-  (  T. 
->  F/ x ps )
52, 4nfbid 1854 . 2  |-  (  T. 
->  F/ x ( ph  <->  ps ) )
65trud 1332 1  |-  F/ x
( ph  <->  ps )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    T. wtru 1325   F/wnf 1553
This theorem is referenced by:  euf  2286  sb8eu  2298  bm1.1  2420  abbi  2545  nfeq  2578  cleqf  2595  sbhypf  2993  ceqsexg  3059  elabgt  3071  elabgf  3072  axrep1  4313  axrep3  4315  axrep4  4316  copsex2t  4435  copsex2g  4436  opelopabsb  4457  opeliunxp2  5005  ralxpf  5011  cbviota  5415  sb8iota  5417  fmptco  5893  nfiso  6036  dfoprab4f  6397  fvopab5  6526  xpf1o  7261  zfcndrep  8481  uzindOLD  10356  gsumcom2  15541  isfildlem  17881  cnextfvval  18088  mbfsup  19548  mbfinf  19549  fmptcof2  24068  subtr2  26309  bnj1468  29154
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554
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