Theorem nfabd2 2921

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2375. (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 2437	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
3	1, 2	nfan 1898	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
4		nfabd2.2	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
5	3, 4	nfabd 2920	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥{𝑦 ∣ 𝜓})
6	5	ex 412	. 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥{𝑦 ∣ 𝜓}))
7		nfab1 2899	. . 3 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜓}
8		eqidd 2735	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑦 ∣ 𝜓} = {𝑦 ∣ 𝜓})
9	8	drnfc1 2917	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥{𝑦 ∣ 𝜓} ↔ Ⅎ𝑦{𝑦 ∣ 𝜓}))
10	7, 9	mpbiri 258	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥{𝑦 ∣ 𝜓})
11	6, 10	pm2.61d2 181	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1537  Ⅎwnf 1782  {cab 2712  Ⅎwnfc 2882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2375  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-nfc 2884

This theorem is referenced by:  nfrab  3462  nfixp  8940