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Theorem nfabd2 2979
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 2005 . . . 4 𝑧(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
2 df-clab 2804 . . . . 5 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
3 nfabd2.1 . . . . . . 7 𝑦𝜑
4 nfnae 2484 . . . . . . 7 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
53, 4nfan 1990 . . . . . 6 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
6 nfabd2.2 . . . . . 6 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
75, 6nfsbd 2607 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
82, 7nfxfrd 1939 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑧 ∈ {𝑦𝜓})
91, 8nfcd 2954 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥{𝑦𝜓})
109ex 399 . 2 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥{𝑦𝜓}))
11 nfab1 2961 . . 3 𝑦{𝑦𝜓}
12 eqidd 2818 . . . 4 (∀𝑥 𝑥 = 𝑦 → {𝑦𝜓} = {𝑦𝜓})
1312drnfc1 2977 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝑥{𝑦𝜓} ↔ 𝑦{𝑦𝜓}))
1411, 13mpbiri 249 . 2 (∀𝑥 𝑥 = 𝑦𝑥{𝑦𝜓})
1510, 14pm2.61d2 173 1 (𝜑𝑥{𝑦𝜓})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1635  wnf 1863  [wsb 2061  wcel 2157  {cab 2803  wnfc 2946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948
This theorem is referenced by:  nfabd  2980  nfrab  3323  nfixp  8174
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