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

Theorem nfiun 3931
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 3907 . 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 2413 . . . 4  |-  F/ y  z  e.  B
52, 4nfrex 2598 . . 3  |-  F/ y E. x  e.  A  z  e.  B
65nfab 2423 . 2  |-  F/_ y { z  |  E. x  e.  A  z  e.  B }
71, 6nfcxfr 2416 1  |-  F/_ y U_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   {cab 2269   F/_wnfc 2406   E.wrex 2544   U_ciun 3905
This theorem is referenced by:  iunab  3948  disjxiun  4020  ovoliunnul  18866  iundisjfi  23363  iundisj2fi  23364  iundisjf  23365  iundisj2f  23366  trpredlem1  24230  trpredrec  24241  bnj1498  29091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-iun 3907
  Copyright terms: Public domain W3C validator