New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  dfnfc2 GIF version

Theorem dfnfc2 3909
 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 2490 . . . 4 (xAxy)
2 id 19 . . . 4 (xAxA)
31, 2nfeqd 2503 . . 3 (xA → Ⅎx y = A)
43alrimiv 1631 . 2 (xAyx y = A)
5 simpr 447 . . . . . 6 ((x A V yx y = A) → yx y = A)
6 df-nfc 2478 . . . . . . 7 (x{A} ↔ yx y {A})
7 elsn 3748 . . . . . . . . 9 (y {A} ↔ y = A)
87nfbii 1569 . . . . . . . 8 (Ⅎx y {A} ↔ Ⅎx y = A)
98albii 1566 . . . . . . 7 (yx y {A} ↔ yx y = A)
106, 9bitri 240 . . . . . 6 (x{A} ↔ yx y = A)
115, 10sylibr 203 . . . . 5 ((x A V yx y = A) → x{A})
1211nfunid 3898 . . . 4 ((x A V yx y = A) → x{A})
13 nfa1 1788 . . . . . 6 xx A V
14 nfnf1 1790 . . . . . . 7 xx y = A
1514nfal 1842 . . . . . 6 xyx y = A
1613, 15nfan 1824 . . . . 5 x(x A V yx y = A)
17 unisng 3908 . . . . . . 7 (A V{A} = A)
1817sps 1754 . . . . . 6 (x A V{A} = A)
1918adantr 451 . . . . 5 ((x A V yx y = A) → {A} = A)
2016, 19nfceqdf 2488 . . . 4 ((x A V yx y = A) → (x{A} ↔ xA))
2112, 20mpbid 201 . . 3 ((x A V yx y = A) → xA)
2221ex 423 . 2 (x A V → (yx y = AxA))
234, 22impbid2 195 1 (x A V → (xAyx y = A))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  {csn 3737  ∪cuni 3891 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator