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Theorem nfabd2 2922
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{𝑦𝜓})

Proof of Theorem nfabd2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1992 . . . 4 𝑧(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
2 df-clab 2747 . . . . 5 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
3 nfabd2.1 . . . . . . 7 𝑦𝜑
4 nfnae 2460 . . . . . . 7 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
53, 4nfan 1977 . . . . . 6 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
6 nfabd2.2 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
75, 6nfsbd 2579 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
82, 7nfxfrd 1929 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑧 ∈ {𝑦𝜓})
91, 8nfcd 2897 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥{𝑦𝜓})
109ex 449 . 2 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥{𝑦𝜓}))
11 nfab1 2904 . . 3 𝑦{𝑦𝜓}
12 eqidd 2761 . . . 4 (∀𝑥 𝑥 = 𝑦 → {𝑦𝜓} = {𝑦𝜓})
1312drnfc1 2920 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝑥{𝑦𝜓} ↔ 𝑦{𝑦𝜓}))
1411, 13mpbiri 248 . 2 (∀𝑥 𝑥 = 𝑦𝑥{𝑦𝜓})
1510, 14pm2.61d2 172 1 (𝜑𝑥{𝑦𝜓})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1630  wnf 1857  [wsb 2046  wcel 2139  {cab 2746  wnfc 2889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891
This theorem is referenced by:  nfabd  2923  nfrab  3262  nfixp  8095
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