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Theorem dfnfc2OLD 4487
Description: Obsolete proof of dfnfc2 4486 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 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem dfnfc2OLD
StepHypRef Expression
1 nfcvd 2794 . . . 4 (𝑥𝐴𝑥𝑦)
2 id 22 . . . 4 (𝑥𝐴𝑥𝐴)
31, 2nfeqd 2801 . . 3 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
43alrimiv 1895 . 2 (𝑥𝐴 → ∀𝑦𝑥 𝑦 = 𝐴)
5 simpr 476 . . . . . 6 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → ∀𝑦𝑥 𝑦 = 𝐴)
6 df-nfc 2782 . . . . . . 7 (𝑥{𝐴} ↔ ∀𝑦𝑥 𝑦 ∈ {𝐴})
7 velsn 4226 . . . . . . . . 9 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
87nfbii 1818 . . . . . . . 8 (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
98albii 1787 . . . . . . 7 (∀𝑦𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦𝑥 𝑦 = 𝐴)
106, 9bitri 264 . . . . . 6 (𝑥{𝐴} ↔ ∀𝑦𝑥 𝑦 = 𝐴)
115, 10sylibr 224 . . . . 5 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → 𝑥{𝐴})
1211nfunid 4475 . . . 4 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → 𝑥 {𝐴})
13 nfa1 2068 . . . . . 6 𝑥𝑥 𝐴𝑉
14 nfnf1 2071 . . . . . . 7 𝑥𝑥 𝑦 = 𝐴
1514nfal 2191 . . . . . 6 𝑥𝑦𝑥 𝑦 = 𝐴
1613, 15nfan 1868 . . . . 5 𝑥(∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴)
17 unisng 4484 . . . . . . 7 (𝐴𝑉 {𝐴} = 𝐴)
1817sps 2093 . . . . . 6 (∀𝑥 𝐴𝑉 {𝐴} = 𝐴)
1918adantr 480 . . . . 5 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → {𝐴} = 𝐴)
2016, 19nfceqdf 2789 . . . 4 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → (𝑥 {𝐴} ↔ 𝑥𝐴))
2112, 20mpbid 222 . . 3 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → 𝑥𝐴)
2221ex 449 . 2 (∀𝑥 𝐴𝑉 → (∀𝑦𝑥 𝑦 = 𝐴𝑥𝐴))
234, 22impbid2 216 1 (∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wnf 1748  wcel 2030  wnfc 2780  {csn 4210   cuni 4468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-un 3612  df-sn 4211  df-pr 4213  df-uni 4469
This theorem is referenced by: (None)
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