Theorem nfabd2 2947

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof shortened by Wolf Lammen, 10-May-2023.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfabd2.1	⊢ Ⅎ𝑦𝜑
nfabd2.2	⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfabd2	⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Proof of Theorem nfabd2

Step	Hyp	Ref	Expression
1		nfabd2.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
2		nfnae 2465	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
3	1, 2	nfan 1932	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
4		nfabd2.2	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
5	3, 4	nfabd 2946	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥{𝑦 ∣ 𝜓})
6	5	ex 418	. 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥{𝑦 ∣ 𝜓}))
7		nfab1 2926	. . 3 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜓}
8		eqidd 2763	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑦 ∣ 𝜓} = {𝑦 ∣ 𝜓})
9	8	drnfc1 2943	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥{𝑦 ∣ 𝜓} ↔ Ⅎ𝑦{𝑦 ∣ 𝜓}))
10	7, 9	mpbiri 261	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥{𝑦 ∣ 𝜓})
11	6, 10	pm2.61d2 183	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1568  Ⅎwnf 1816  {cab 2740  Ⅎwnfc 2909

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-13 2403  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-nfc 2911

This theorem is used by:  nfrab  3451  nfixp  8928