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Theorem nfbid 1903
 Description: If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 ↔ 𝜒). (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 478 . 2 ((𝜓𝜒) ↔ ((𝜓𝜒) ∧ (𝜒𝜓)))
2 nfbid.1 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
3 nfbid.2 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
42, 3nfimd 1895 . . 3 (𝜑 → Ⅎ𝑥(𝜓𝜒))
53, 2nfimd 1895 . . 3 (𝜑 → Ⅎ𝑥(𝜒𝜓))
64, 5nfand 1898 . 2 (𝜑 → Ⅎ𝑥((𝜓𝜒) ∧ (𝜒𝜓)))
71, 6nfxfrd 1855 1 (𝜑 → Ⅎ𝑥(𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786 This theorem is referenced by:  nfbi  1904  nfeqd  2929  nfiotadw  6297  nfiotad  6299  iota2df  6322  axextnd  10051  axrepndlem1  10052  axrepndlem2  10053  axacndlem4  10070  axacndlem5  10071  axacnd  10072  axextdist  33291  copsex2d  34834  cbveud  35069  wl-eudf  35253  wl-sb8eut  35258
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