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Theorem nfabd2 2507
 Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
nfabd2.1 yφ
nfabd2.2 ((φ ¬ x x = y) → Ⅎxψ)
Assertion
Ref Expression
nfabd2 (φx{y ψ})

Proof of Theorem nfabd2
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . 4 z(φ ¬ x x = y)
2 df-clab 2340 . . . . 5 (z {y ψ} ↔ [z / y]ψ)
3 nfabd2.1 . . . . . . 7 yφ
4 nfnae 1956 . . . . . . 7 y ¬ x x = y
53, 4nfan 1824 . . . . . 6 y(φ ¬ x x = y)
6 nfabd2.2 . . . . . 6 ((φ ¬ x x = y) → Ⅎxψ)
75, 6nfsbd 2111 . . . . 5 ((φ ¬ x x = y) → Ⅎx[z / y]ψ)
82, 7nfxfrd 1571 . . . 4 ((φ ¬ x x = y) → Ⅎx z {y ψ})
91, 8nfcd 2484 . . 3 ((φ ¬ x x = y) → x{y ψ})
109ex 423 . 2 (φ → (¬ x x = yx{y ψ}))
11 nfab1 2491 . . 3 y{y ψ}
12 eqidd 2354 . . . 4 (x x = y → {y ψ} = {y ψ})
1312drnfc1 2505 . . 3 (x x = y → (x{y ψ} ↔ y{y ψ}))
1411, 13mpbiri 224 . 2 (x x = yx{y ψ})
1510, 14pm2.61d2 152 1 (φx{y ψ})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476 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-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 This theorem is referenced by:  nfabd  2508  nfrab  2792
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