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Theorem nfunid 3958
Description: Deduction version of nfuni 3957. (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 3953 . 2  |-  U. A  =  { y  |  E. z  e.  A  y  e.  z }
2 nfv 1626 . . 3  |-  F/ y
ph
3 nfv 1626 . . . 4  |-  F/ z
ph
4 nfunid.3 . . . 4  |-  ( ph  -> 
F/_ x A )
5 nfvd 1627 . . . 4  |-  ( ph  ->  F/ x  y  e.  z )
63, 4, 5nfrexd 2695 . . 3  |-  ( ph  ->  F/ x E. z  e.  A  y  e.  z )
72, 6nfabd 2536 . 2  |-  ( ph  -> 
F/_ x { y  |  E. z  e.  A  y  e.  z } )
81, 7nfcxfrd 2515 1  |-  ( ph  -> 
F/_ x U. A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   {cab 2367   F/_wnfc 2504   E.wrex 2644   U.cuni 3951
This theorem is referenced by:  dfnfc2  3969  nfiotad  5355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ral 2648  df-rex 2649  df-uni 3952
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