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Theorem nfif 3686
 Description: Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Hypotheses
Ref Expression
nfif.1 xφ
nfif.2 xA
nfif.3 xB
Assertion
Ref Expression
nfif x if(φ, A, B)

Proof of Theorem nfif
StepHypRef Expression
1 nfif.1 . . . 4 xφ
21a1i 10 . . 3 ( ⊤ → Ⅎxφ)
3 nfif.2 . . . 4 xA
43a1i 10 . . 3 ( ⊤ → xA)
5 nfif.3 . . . 4 xB
65a1i 10 . . 3 ( ⊤ → xB)
72, 4, 6nfifd 3685 . 2 ( ⊤ → x if(φ, A, B))
87trud 1323 1 x if(φ, A, B)
 Colors of variables: wff setvar class Syntax hints:   ⊤ wtru 1316  Ⅎwnf 1544  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:  csbifg  3690
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