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Theorem abexv 4324
 Description: When x does not occur in φ, {x ∣ φ} is a set. (Contributed by SF, 17-Jan-2015.)
Assertion
Ref Expression
abexv {x φ} V
Distinct variable group:   φ,x

Proof of Theorem abexv
StepHypRef Expression
1 abvor0 3567 . 2 ({x φ} = V {x φ} = )
2 vvex 4109 . . . 4 V V
3 eleq1 2413 . . . 4 ({x φ} = V → ({x φ} V ↔ V V))
42, 3mpbiri 224 . . 3 ({x φ} = V → {x φ} V)
5 0ex 4110 . . . 4 V
6 eleq1 2413 . . . 4 ({x φ} = → ({x φ} V ↔ V))
75, 6mpbiri 224 . . 3 ({x φ} = → {x φ} V)
84, 7jaoi 368 . 2 (({x φ} = V {x φ} = ) → {x φ} V)
91, 8ax-mp 5 1 {x φ} V
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859  ∅c0 3550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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  ax-nin 4078 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by:  nncaddccl  4419  preaddccan2lem1  4454  ltfintrilem1  4465  leconnnc  6218  addccan2nclem2  6264  nchoicelem16  6304
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