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Theorem nfunid 3655
Description: Deduction version of nfuni 3654. (Contributed by NM, 18-Feb-2013.)
Hypothesis
Ref Expression
nfunid.3  |-  ( ph  -> 
F/_ x A )
Assertion
Ref Expression
nfunid  |-  ( ph  -> 
F/_ x U. A
)

Proof of Theorem nfunid
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfuni2 3650 . 2  |-  U. A  =  { y  |  E. z  e.  A  y  e.  z }
2 nfv 1466 . . 3  |-  F/ y
ph
3 nfv 1466 . . . 4  |-  F/ z
ph
4 nfunid.3 . . . 4  |-  ( ph  -> 
F/_ x A )
5 nfvd 1467 . . . 4  |-  ( ph  ->  F/ x  y  e.  z )
63, 4, 5nfrexdxy 2411 . . 3  |-  ( ph  ->  F/ x E. z  e.  A  y  e.  z )
72, 6nfabd 2247 . 2  |-  ( ph  -> 
F/_ x { y  |  E. z  e.  A  y  e.  z } )
81, 7nfcxfrd 2226 1  |-  ( ph  -> 
F/_ x U. A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   {cab 2074   F/_wnfc 2215   E.wrex 2360   U.cuni 3648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-uni 3649
This theorem is referenced by:  dfnfc2  3666  nfiotadxy  4970
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