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

Theorem issetf 2540
Description: A version of isset that does not require x and A to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.)
Hypothesis
Ref Expression
issetf.1 xA
Assertion
Ref Expression
issetf (A V ↔ x x = A)

Proof of Theorem issetf
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 isset 2539 . 2 (A V ↔ y y = A)
2 issetf.1 . . . 4 xA
32nfeq2 2171 . . 3 x y = A
4 nfv 1402 . . 3 y x = A
5 eqeq1 2028 . . 3 (y = x → (y = Ax = A))
63, 4, 5cbvex 1621 . 2 (y y = Ax x = A)
71, 6bitri 173 1 (A V ↔ x x = A)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1228  wex 1362   wcel 1374  wnfc 2147  Vcvv 2535
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-v 2537
This theorem is referenced by:  vtoclgf  2589  spcimgft  2606  spcimegft  2608  bj-vtoclgft  7021
  Copyright terms: Public domain W3C validator