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Theorem nfbid 1897
Description: If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)
Hypotheses
Ref Expression
nfbid.1 (𝜑 → Ⅎ𝑥𝜓)
nfbid.2 (𝜑 → Ⅎ𝑥𝜒)
Assertion
Ref Expression
nfbid (𝜑 → Ⅎ𝑥(𝜓𝜒))

Proof of Theorem nfbid
StepHypRef Expression
1 dfbi2 473 . 2 ((𝜓𝜒) ↔ ((𝜓𝜒) ∧ (𝜒𝜓)))
2 nfbid.1 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
3 nfbid.2 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
42, 3nfimd 1889 . . 3 (𝜑 → Ⅎ𝑥(𝜓𝜒))
53, 2nfimd 1889 . . 3 (𝜑 → Ⅎ𝑥(𝜒𝜓))
64, 5nfand 1892 . 2 (𝜑 → Ⅎ𝑥((𝜓𝜒) ∧ (𝜒𝜓)))
71, 6nfxfrd 1848 1 (𝜑 → Ⅎ𝑥(𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778
This theorem is referenced by:  nfbi  1898  nfeqd  2902  nfiotadw  6504  nfiotad  6506  iota2df  6536  axextnd  10616  axrepndlem1  10617  axrepndlem2  10618  axacndlem4  10635  axacndlem5  10636  axacnd  10637  axextdist  35526  copsex2d  36749  cbveud  36982  wl-eudf  37170  wl-sb8eut  37176  wl-sb8eutv  37177
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