Theorem dfnfc2OLD 4391

Description: Obsolete proof of dfnfc2 4390 as of 26-Jul-2021. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dfnfc2OLD	⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem dfnfc2OLD

Step	Hyp	Ref	Expression
1		nfcvd 2752	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦)
2		id 22	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴)
3	1, 2	nfeqd 2758	. . 3 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
4	3	alrimiv 1842	. 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
5		simpr 476	. . . . . 6 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
6		df-nfc 2740	. . . . . . 7 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴})
7		velsn 4141	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
8	7	nfbii 1770	. . . . . . . 8 ⊢ (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
9	8	albii 1737	. . . . . . 7 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
10	6, 9	bitri 263	. . . . . 6 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
11	5, 10	sylibr 223	. . . . 5 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → Ⅎ𝑥{𝐴})
12	11	nfunid 4379	. . . 4 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → Ⅎ𝑥∪ {𝐴})
13		nfa1 2015	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 𝐴 ∈ 𝑉
14		nfnf1 2018	. . . . . . 7 ⊢ Ⅎ𝑥Ⅎ𝑥 𝑦 = 𝐴
15	14	nfal 2139	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥 𝑦 = 𝐴
16	13, 15	nfan 1816	. . . . 5 ⊢ Ⅎ𝑥(∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
17		unisng 4388	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
18	17	sps 2043	. . . . . 6 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
19	18	adantr 480	. . . . 5 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → ∪ {𝐴} = 𝐴)
20	16, 19	nfceqdf 2747	. . . 4 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → (Ⅎ𝑥∪ {𝐴} ↔ Ⅎ𝑥𝐴))
21	12, 20	mpbid 221	. . 3 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → Ⅎ𝑥𝐴)
22	21	ex 449	. 2 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴))
23	4, 22	impbid2 215	1 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  {csn 4125  ∪ cuni 4372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128  df-uni 4373

This theorem is referenced by: (None)