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Theorem nfreud 2872
Description: Deduction version of nfreu 2874. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
nfreud.1  |-  F/ y
ph
nfreud.2  |-  ( ph  -> 
F/_ x A )
nfreud.3  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfreud  |-  ( ph  ->  F/ x E! y  e.  A  ps )

Proof of Theorem nfreud
StepHypRef Expression
1 df-reu 2704 . 2  |-  ( E! y  e.  A  ps  <->  E! y ( y  e.  A  /\  ps )
)
2 nfreud.1 . . 3  |-  F/ y
ph
3 nfcvf 2593 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
43adantl 453 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x y )
5 nfreud.2 . . . . . 6  |-  ( ph  -> 
F/_ x A )
65adantr 452 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x A )
74, 6nfeld 2586 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  y  e.  A )
8 nfreud.3 . . . . 5  |-  ( ph  ->  F/ x ps )
98adantr 452 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
107, 9nfand 1843 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ( y  e.  A  /\  ps ) )
112, 10nfeud2 2292 . 2  |-  ( ph  ->  F/ x E! y ( y  e.  A  /\  ps ) )
121, 11nfxfrd 1580 1  |-  ( ph  ->  F/ x E! y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1549   F/wnf 1553    e. wcel 1725   E!weu 2280   F/_wnfc 2558   E!wreu 2699
This theorem is referenced by:  nfreu  2874  nfriotad  6550
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 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-eu 2284  df-cleq 2428  df-clel 2431  df-nfc 2560  df-reu 2704
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