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Theorem nfunid 4242
Description: Deduction version of nfuni 4241. (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 4237 . 2  |-  U. A  =  { y  |  E. z  e.  A  y  e.  z }
2 nfv 1712 . . 3  |-  F/ y
ph
3 nfv 1712 . . . 4  |-  F/ z
ph
4 nfunid.3 . . . 4  |-  ( ph  -> 
F/_ x A )
5 nfvd 1713 . . . 4  |-  ( ph  ->  F/ x  y  e.  z )
63, 4, 5nfrexd 2916 . . 3  |-  ( ph  ->  F/ x E. z  e.  A  y  e.  z )
72, 6nfabd 2638 . 2  |-  ( ph  -> 
F/_ x { y  |  E. z  e.  A  y  e.  z } )
81, 7nfcxfrd 2615 1  |-  ( ph  -> 
F/_ x U. A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   {cab 2439   F/_wnfc 2602   E.wrex 2805   U.cuni 4235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-uni 4236
This theorem is referenced by:  dfnfc2  4253  nfiotad  5537
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