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Theorem dfnfc2 3814
Description: An alternate statement of the effective freeness of a class 𝐴, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
dfnfc2 (∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem dfnfc2
StepHypRef Expression
1 nfcvd 2313 . . . 4 (𝑥𝐴𝑥𝑦)
2 id 19 . . . 4 (𝑥𝐴𝑥𝐴)
31, 2nfeqd 2327 . . 3 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
43alrimiv 1867 . 2 (𝑥𝐴 → ∀𝑦𝑥 𝑦 = 𝐴)
5 simpr 109 . . . . . 6 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → ∀𝑦𝑥 𝑦 = 𝐴)
6 df-nfc 2301 . . . . . . 7 (𝑥{𝐴} ↔ ∀𝑦𝑥 𝑦 ∈ {𝐴})
7 velsn 3600 . . . . . . . . 9 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
87nfbii 1466 . . . . . . . 8 (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
98albii 1463 . . . . . . 7 (∀𝑦𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦𝑥 𝑦 = 𝐴)
106, 9bitri 183 . . . . . 6 (𝑥{𝐴} ↔ ∀𝑦𝑥 𝑦 = 𝐴)
115, 10sylibr 133 . . . . 5 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → 𝑥{𝐴})
1211nfunid 3803 . . . 4 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → 𝑥 {𝐴})
13 nfa1 1534 . . . . . 6 𝑥𝑥 𝐴𝑉
14 nfnf1 1537 . . . . . . 7 𝑥𝑥 𝑦 = 𝐴
1514nfal 1569 . . . . . 6 𝑥𝑦𝑥 𝑦 = 𝐴
1613, 15nfan 1558 . . . . 5 𝑥(∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴)
17 unisng 3813 . . . . . . 7 (𝐴𝑉 {𝐴} = 𝐴)
1817sps 1530 . . . . . 6 (∀𝑥 𝐴𝑉 {𝐴} = 𝐴)
1918adantr 274 . . . . 5 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → {𝐴} = 𝐴)
2016, 19nfceqdf 2311 . . . 4 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → (𝑥 {𝐴} ↔ 𝑥𝐴))
2112, 20mpbid 146 . . 3 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → 𝑥𝐴)
2221ex 114 . 2 (∀𝑥 𝐴𝑉 → (∀𝑦𝑥 𝑦 = 𝐴𝑥𝐴))
234, 22impbid2 142 1 (∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1346   = wceq 1348  wnf 1453  wcel 2141  wnfc 2299  {csn 3583   cuni 3796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797
This theorem is referenced by:  eusv2nf  4441
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