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

Theorem nfiinya 3812
Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiunya.1  |-  F/_ y A
nfiunya.2  |-  F/_ y B
Assertion
Ref Expression
nfiinya  |-  F/_ y |^|_ x  e.  A  B
Distinct variable group:    x, A
Allowed substitution hints:    A( y)    B( x, y)

Proof of Theorem nfiinya
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-iin 3786 . 2  |-  |^|_ x  e.  A  B  =  { z  |  A. x  e.  A  z  e.  B }
2 nfiunya.1 . . . 4  |-  F/_ y A
3 nfiunya.2 . . . . 5  |-  F/_ y B
43nfcri 2252 . . . 4  |-  F/ y  z  e.  B
52, 4nfralya 2450 . . 3  |-  F/ y A. x  e.  A  z  e.  B
65nfab 2263 . 2  |-  F/_ y { z  |  A. x  e.  A  z  e.  B }
71, 6nfcxfr 2255 1  |-  F/_ y |^|_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1465   {cab 2103   F/_wnfc 2245   A.wral 2393   |^|_ciin 3784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-iin 3786
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator