ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eusvnf Structured version   GIF version

Theorem eusvnf 4131
Description: Even if x is free in A, it is effectively bound when A(x) is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
eusvnf (∃!yx y = AxA)
Distinct variable groups:   x,y   y,A
Allowed substitution hint:   A(x)

Proof of Theorem eusvnf
Dummy variables z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 euex 1908 . 2 (∃!yx y = Ayx y = A)
2 vex 2534 . . . . . . 7 z V
3 nfcv 2156 . . . . . . . 8 xz
4 nfcsb1v 2855 . . . . . . . . 9 xz / xA
54nfeq2 2167 . . . . . . . 8 x y = z / xA
6 csbeq1a 2833 . . . . . . . . 9 (x = zA = z / xA)
76eqeq2d 2029 . . . . . . . 8 (x = z → (y = Ay = z / xA))
83, 5, 7spcgf 2608 . . . . . . 7 (z V → (x y = Ay = z / xA))
92, 8ax-mp 7 . . . . . 6 (x y = Ay = z / xA)
10 vex 2534 . . . . . . 7 w V
11 nfcv 2156 . . . . . . . 8 xw
12 nfcsb1v 2855 . . . . . . . . 9 xw / xA
1312nfeq2 2167 . . . . . . . 8 x y = w / xA
14 csbeq1a 2833 . . . . . . . . 9 (x = wA = w / xA)
1514eqeq2d 2029 . . . . . . . 8 (x = w → (y = Ay = w / xA))
1611, 13, 15spcgf 2608 . . . . . . 7 (w V → (x y = Ay = w / xA))
1710, 16ax-mp 7 . . . . . 6 (x y = Ay = w / xA)
189, 17eqtr3d 2052 . . . . 5 (x y = Az / xA = w / xA)
1918alrimivv 1733 . . . 4 (x y = Azwz / xA = w / xA)
20 sbnfc2 2879 . . . 4 (xAzwz / xA = w / xA)
2119, 20sylibr 137 . . 3 (x y = AxA)
2221exlimiv 1467 . 2 (yx y = AxA)
231, 22syl 14 1 (∃!yx y = AxA)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1224   = wceq 1226  wex 1358   wcel 1370  ∃!weu 1878  wnfc 2143  Vcvv 2531  csb 2825
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-sbc 2738  df-csb 2826
This theorem is referenced by:  eusvnfb  4132  eusv2i  4133
  Copyright terms: Public domain W3C validator