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Theorem dfnfc2 3589
Description: An alternative 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 A 𝑉 → (xAyx 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 2176 . . . 4 (xAxy)
2 id 19 . . . 4 (xAxA)
31, 2nfeqd 2189 . . 3 (xA → Ⅎx y = A)
43alrimiv 1751 . 2 (xAyx y = A)
5 simpr 103 . . . . . 6 ((x A 𝑉 yx y = A) → yx y = A)
6 df-nfc 2164 . . . . . . 7 (x{A} ↔ yx y {A})
7 elsn 3382 . . . . . . . . 9 (y {A} ↔ y = A)
87nfbii 1359 . . . . . . . 8 (Ⅎx y {A} ↔ Ⅎx y = A)
98albii 1356 . . . . . . 7 (yx y {A} ↔ yx y = A)
106, 9bitri 173 . . . . . 6 (x{A} ↔ yx y = A)
115, 10sylibr 137 . . . . 5 ((x A 𝑉 yx y = A) → x{A})
1211nfunid 3578 . . . 4 ((x A 𝑉 yx y = A) → x {A})
13 nfa1 1431 . . . . . 6 xx A 𝑉
14 nfnf1 1433 . . . . . . 7 xx y = A
1514nfal 1465 . . . . . 6 xyx y = A
1613, 15nfan 1454 . . . . 5 x(x A 𝑉 yx y = A)
17 unisng 3588 . . . . . . 7 (A 𝑉 {A} = A)
1817sps 1427 . . . . . 6 (x A 𝑉 {A} = A)
1918adantr 261 . . . . 5 ((x A 𝑉 yx y = A) → {A} = A)
2016, 19nfceqdf 2174 . . . 4 ((x A 𝑉 yx y = A) → (x {A} ↔ xA))
2112, 20mpbid 135 . . 3 ((x A 𝑉 yx y = A) → xA)
2221ex 108 . 2 (x A 𝑉 → (yx y = AxA))
234, 22impbid2 131 1 (x A 𝑉 → (xAyx y = A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wnf 1346   wcel 1390  wnfc 2162  {csn 3367   cuni 3571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572
This theorem is referenced by:  eusv2nf  4154
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