ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfin Unicode version

Theorem nfin 3356
Description: Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfin.1  |-  F/_ x A
nfin.2  |-  F/_ x B
Assertion
Ref Expression
nfin  |-  F/_ x
( A  i^i  B
)

Proof of Theorem nfin
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfin5 3151 . 2  |-  ( A  i^i  B )  =  { y  e.  A  |  y  e.  B }
2 nfin.2 . . . 4  |-  F/_ x B
32nfcri 2326 . . 3  |-  F/ x  y  e.  B
4 nfin.1 . . 3  |-  F/_ x A
53, 4nfrabxy 2671 . 2  |-  F/_ x { y  e.  A  |  y  e.  B }
61, 5nfcxfr 2329 1  |-  F/_ x
( A  i^i  B
)
Colors of variables: wff set class
Syntax hints:    e. wcel 2160   F/_wnfc 2319   {crab 2472    i^i cin 3143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-in 3150
This theorem is referenced by:  csbing  3357  nfres  4927
  Copyright terms: Public domain W3C validator