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

Theorem intiin 3702
 Description: Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
Assertion
Ref Expression
intiin A = x A x
Distinct variable group:   x,A

Proof of Theorem intiin
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfint2 3608 . 2 A = {yx A y x}
2 df-iin 3651 . 2 x A x = {yx A y x}
31, 2eqtr4i 2060 1 A = x A x
 Colors of variables: wff set class Syntax hints:   = wceq 1242  {cab 2023  ∀wral 2300  ∩ cint 3606  ∩ ciin 3649 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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  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-ral 2305  df-int 3607  df-iin 3651 This theorem is referenced by:  relint  4404
 Copyright terms: Public domain W3C validator