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Theorem dfnfc2 3572
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 2161 . . . 4 (xAxy)
2 id 19 . . . 4 (xAxA)
31, 2nfeqd 2174 . . 3 (xA → Ⅎx y = A)
43alrimiv 1736 . 2 (xAyx y = A)
5 simpr 103 . . . . . 6 ((x A 𝑉 yx y = A) → yx y = A)
6 df-nfc 2149 . . . . . . 7 (x{A} ↔ yx y {A})
7 elsn 3365 . . . . . . . . 9 (y {A} ↔ y = A)
87nfbii 1342 . . . . . . . 8 (Ⅎx y {A} ↔ Ⅎx y = A)
98albii 1339 . . . . . . 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 3561 . . . 4 ((x A 𝑉 yx y = A) → x {A})
13 nfa1 1416 . . . . . 6 xx A 𝑉
14 nfnf1 1418 . . . . . . 7 xx y = A
1514nfal 1450 . . . . . 6 xyx y = A
1613, 15nfan 1439 . . . . 5 x(x A 𝑉 yx y = A)
17 unisng 3571 . . . . . . 7 (A 𝑉 {A} = A)
1817sps 1412 . . . . . 6 (x A 𝑉 {A} = A)
1918adantr 261 . . . . 5 ((x A 𝑉 yx y = A) → {A} = A)
2016, 19nfceqdf 2159 . . . 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 1226   = wceq 1228  wnf 1329   wcel 1374  wnfc 2147  {csn 3350   cuni 3554
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357  df-uni 3555
This theorem is referenced by:  eusv2nf  4138
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