MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfiun Structured version   Unicode version

Theorem nfiun 4119
Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiun.1  |-  F/_ y A
nfiun.2  |-  F/_ y B
Assertion
Ref Expression
nfiun  |-  F/_ y U_ x  e.  A  B

Proof of Theorem nfiun
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-iun 4095 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
2 nfiun.1 . . . 4  |-  F/_ y A
3 nfiun.2 . . . . 5  |-  F/_ y B
43nfcri 2566 . . . 4  |-  F/ y  z  e.  B
52, 4nfrex 2761 . . 3  |-  F/ y E. x  e.  A  z  e.  B
65nfab 2576 . 2  |-  F/_ y { z  |  E. x  e.  A  z  e.  B }
71, 6nfcxfr 2569 1  |-  F/_ y U_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   {cab 2422   F/_wnfc 2559   E.wrex 2706   U_ciun 4093
This theorem is referenced by:  iunab  4137  disjxiun  4209  ovoliunnul  19403  iundisjf  24029  iundisj2f  24030  iundisjfi  24152  iundisj2fi  24153  trpredlem1  25505  trpredrec  25516  bnj1498  29430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-iun 4095
  Copyright terms: Public domain W3C validator