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Theorem nfabd2 2926
Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2367. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof shortened by Wolf Lammen, 10-May-2023.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfabd2.1 𝑦𝜑
nfabd2.2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfabd2 (𝜑𝑥{𝑦𝜓})

Proof of Theorem nfabd2
StepHypRef Expression
1 nfabd2.1 . . . . 5 𝑦𝜑
2 nfnae 2429 . . . . 5 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
31, 2nfan 1895 . . . 4 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
4 nfabd2.2 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
53, 4nfabd 2925 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥{𝑦𝜓})
65ex 412 . 2 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥{𝑦𝜓}))
7 nfab1 2901 . . 3 𝑦{𝑦𝜓}
8 eqidd 2729 . . . 4 (∀𝑥 𝑥 = 𝑦 → {𝑦𝜓} = {𝑦𝜓})
98drnfc1 2919 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝑥{𝑦𝜓} ↔ 𝑦{𝑦𝜓}))
107, 9mpbiri 258 . 2 (∀𝑥 𝑥 = 𝑦𝑥{𝑦𝜓})
116, 10pm2.61d2 181 1 (𝜑𝑥{𝑦𝜓})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1532  wnf 1778  {cab 2705  wnfc 2879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2367  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-nfc 2881
This theorem is referenced by:  nfrab  3469  nfixp  8936
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