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Theorem nfunid 4015
Description: Deduction version of nfuni 4014. (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 4010 . 2  |-  U. A  =  { y  |  E. z  e.  A  y  e.  z }
2 nfv 1629 . . 3  |-  F/ y
ph
3 nfv 1629 . . . 4  |-  F/ z
ph
4 nfunid.3 . . . 4  |-  ( ph  -> 
F/_ x A )
5 nfvd 1630 . . . 4  |-  ( ph  ->  F/ x  y  e.  z )
63, 4, 5nfrexd 2751 . . 3  |-  ( ph  ->  F/ x E. z  e.  A  y  e.  z )
72, 6nfabd 2591 . 2  |-  ( ph  -> 
F/_ x { y  |  E. z  e.  A  y  e.  z } )
81, 7nfcxfrd 2570 1  |-  ( ph  -> 
F/_ x U. A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   {cab 2422   F/_wnfc 2559   E.wrex 2699   U.cuni 4008
This theorem is referenced by:  dfnfc2  4026  nfiotad  5414
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-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 2703  df-rex 2704  df-uni 4009
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