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Theorem nfabd2 3007
 Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof shortened by Wolf Lammen, 10-May-2023.)
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 2453 . . . . 5 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
31, 2nfan 1893 . . . 4 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
4 nfabd2.2 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
53, 4nfabd 3006 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥{𝑦𝜓})
65ex 413 . 2 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥{𝑦𝜓}))
7 nfab1 2984 . . 3 𝑦{𝑦𝜓}
8 eqidd 2827 . . . 4 (∀𝑥 𝑥 = 𝑦 → {𝑦𝜓} = {𝑦𝜓})
98drnfc1 3002 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝑥{𝑦𝜓} ↔ 𝑦{𝑦𝜓}))
107, 9mpbiri 259 . 2 (∀𝑥 𝑥 = 𝑦𝑥{𝑦𝜓})
116, 10pm2.61d2 182 1 (𝜑𝑥{𝑦𝜓})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1528  Ⅎwnf 1777  {cab 2804  Ⅎwnfc 2966 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968 This theorem is referenced by:  nfabdOLD  3009  nfrab  3392  nfixp  8470
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