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Theorem nfifd 3685
 Description: Deduction version of nfif 3686. (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 (φx if(ψ, A, B))

Proof of Theorem nfifd
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfif2 3664 . 2 if(ψ, A, B) = {y ((y Bψ) → (y A ψ))}
2 nfv 1619 . . 3 yφ
3 nfifd.4 . . . . . 6 (φxB)
43nfcrd 2502 . . . . 5 (φ → Ⅎx y B)
5 nfifd.2 . . . . 5 (φ → Ⅎxψ)
64, 5nfimd 1808 . . . 4 (φ → Ⅎx(y Bψ))
7 nfifd.3 . . . . . 6 (φxA)
87nfcrd 2502 . . . . 5 (φ → Ⅎx y A)
98, 5nfand 1822 . . . 4 (φ → Ⅎx(y A ψ))
106, 9nfimd 1808 . . 3 (φ → Ⅎx((y Bψ) → (y A ψ)))
112, 10nfabd 2508 . 2 (φx{y ((y Bψ) → (y A ψ))})
121, 11nfcxfrd 2487 1 (φx if(ψ, A, B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  Ⅎwnf 1544   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476   ifcif 3662 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-if 3663 This theorem is referenced by:  nfif  3686
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