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Theorem nfrab 2792
 Description: A variable not free in a wff remains so in a restricted class abstraction. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.)
Hypotheses
Ref Expression
nfrab.1 xφ
nfrab.2 xA
Assertion
Ref Expression
nfrab x{y A φ}

Proof of Theorem nfrab
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-rab 2623 . 2 {y A φ} = {y (y A φ)}
2 nftru 1554 . . . 4 y
3 nfrab.2 . . . . . . . 8 xA
43nfcri 2483 . . . . . . 7 x z A
5 eleq1 2413 . . . . . . 7 (z = y → (z Ay A))
64, 5dvelimnf 2017 . . . . . 6 x x = y → Ⅎx y A)
7 nfrab.1 . . . . . . 7 xφ
87a1i 10 . . . . . 6 x x = y → Ⅎxφ)
96, 8nfand 1822 . . . . 5 x x = y → Ⅎx(y A φ))
109adantl 452 . . . 4 (( ⊤ ¬ x x = y) → Ⅎx(y A φ))
112, 10nfabd2 2507 . . 3 ( ⊤ → x{y (y A φ)})
1211trud 1323 . 2 x{y (y A φ)}
131, 12nfcxfr 2486 1 x{y A φ}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 358   ⊤ wtru 1316  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476  {crab 2618 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-rab 2623 This theorem is referenced by: (None)
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