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Theorem nfifd 3349
Description: Deduction version of nfif 3350. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
nfifd.2 (φ → Ⅎxψ)
nfifd.3 (φxA)
nfifd.4 (φxB)
Assertion
Ref Expression
nfifd (φxif(ψ, A, B))

Proof of Theorem nfifd
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-if 3326 . 2 if(ψ, A, B) = {y ∣ ((y A ψ) (y B ¬ ψ))}
2 nfv 1418 . . 3 yφ
3 nfifd.3 . . . . . 6 (φxA)
43nfcrd 2188 . . . . 5 (φ → Ⅎx y A)
5 nfifd.2 . . . . 5 (φ → Ⅎxψ)
64, 5nfand 1457 . . . 4 (φ → Ⅎx(y A ψ))
7 nfifd.4 . . . . . 6 (φxB)
87nfcrd 2188 . . . . 5 (φ → Ⅎx y B)
95nfnd 1544 . . . . 5 (φ → Ⅎx ¬ ψ)
108, 9nfand 1457 . . . 4 (φ → Ⅎx(y B ¬ ψ))
116, 10nford 1456 . . 3 (φ → Ⅎx((y A ψ) (y B ¬ ψ)))
122, 11nfabd 2193 . 2 (φx{y ∣ ((y A ψ) (y B ¬ ψ))})
131, 12nfcxfrd 2173 1 (φxif(ψ, A, B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 628  wnf 1346   wcel 1390  {cab 2023  wnfc 2162  ifcif 3325
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-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-if 3326
This theorem is referenced by:  nfif  3350
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