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Theorem nfreud 2712
Description: Deduction version of nfreu 2714. (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 2550 . 2  |-  ( E! y  e.  A  ps  <->  E! y ( y  e.  A  /\  ps )
)
2 nfreud.1 . . 3  |-  F/ y
ph
3 nfcvf 2441 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
43adantl 452 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x y )
5 nfreud.2 . . . . . 6  |-  ( ph  -> 
F/_ x A )
65adantr 451 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x A )
74, 6nfeld 2434 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  y  e.  A )
8 nfreud.3 . . . . 5  |-  ( ph  ->  F/ x ps )
98adantr 451 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
107, 9nfand 1763 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ( y  e.  A  /\  ps ) )
112, 10nfeud2 2155 . 2  |-  ( ph  ->  F/ x E! y ( y  e.  A  /\  ps ) )
121, 11nfxfrd 1558 1  |-  ( ph  ->  F/ x E! y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527   F/wnf 1531    = wceq 1623    e. wcel 1684   E!weu 2143   F/_wnfc 2406   E!wreu 2545
This theorem is referenced by:  nfreu  2714  nfriotad  6313
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-eu 2147  df-cleq 2276  df-clel 2279  df-nfc 2408  df-reu 2550
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