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Theorem dfnfc2OLD 4421
 Description: Obsolete proof of dfnfc2 4420 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 2762 . . . 4 (𝑥𝐴𝑥𝑦)
2 id 22 . . . 4 (𝑥𝐴𝑥𝐴)
31, 2nfeqd 2768 . . 3 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
43alrimiv 1852 . 2 (𝑥𝐴 → ∀𝑦𝑥 𝑦 = 𝐴)
5 simpr 477 . . . . . 6 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → ∀𝑦𝑥 𝑦 = 𝐴)
6 df-nfc 2750 . . . . . . 7 (𝑥{𝐴} ↔ ∀𝑦𝑥 𝑦 ∈ {𝐴})
7 velsn 4164 . . . . . . . . 9 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
87nfbii 1775 . . . . . . . 8 (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
98albii 1744 . . . . . . 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 4409 . . . 4 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → 𝑥 {𝐴})
13 nfa1 2025 . . . . . 6 𝑥𝑥 𝐴𝑉
14 nfnf1 2028 . . . . . . 7 𝑥𝑥 𝑦 = 𝐴
1514nfal 2150 . . . . . 6 𝑥𝑦𝑥 𝑦 = 𝐴
1613, 15nfan 1825 . . . . 5 𝑥(∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴)
17 unisng 4418 . . . . . . 7 (𝐴𝑉 {𝐴} = 𝐴)
1817sps 2053 . . . . . 6 (∀𝑥 𝐴𝑉 {𝐴} = 𝐴)
1918adantr 481 . . . . 5 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → {𝐴} = 𝐴)
2016, 19nfceqdf 2757 . . . 4 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → (𝑥 {𝐴} ↔ 𝑥𝐴))
2112, 20mpbid 222 . . 3 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → 𝑥𝐴)
2221ex 450 . 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 384  ∀wal 1478   = wceq 1480  Ⅎwnf 1705   ∈ wcel 1987  Ⅎwnfc 2748  {csn 4148  ∪ cuni 4402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-un 3560  df-sn 4149  df-pr 4151  df-uni 4403 This theorem is referenced by: (None)
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