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Theorem nfunid 3834
Description: Deduction version of nfuni 3833. (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 3829 . 2  |-  U. A  =  { y  |  E. z  e.  A  y  e.  z }
2 nfv 1605 . . 3  |-  F/ y
ph
3 nfv 1605 . . . 4  |-  F/ z
ph
4 nfunid.3 . . . 4  |-  ( ph  -> 
F/_ x A )
5 nfvd 1606 . . . 4  |-  ( ph  ->  F/ x  y  e.  z )
63, 4, 5nfrexd 2595 . . 3  |-  ( ph  ->  F/ x E. z  e.  A  y  e.  z )
72, 6nfabd 2438 . 2  |-  ( ph  -> 
F/_ x { y  |  E. z  e.  A  y  e.  z } )
81, 7nfcxfrd 2417 1  |-  ( ph  -> 
F/_ x U. A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   {cab 2269   F/_wnfc 2406   E.wrex 2544   U.cuni 3827
This theorem is referenced by:  dfnfc2  3845  nfiotad  5222
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-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-uni 3828
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