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Theorem nfrmod 2726
Description: Deduction version of nfrmo 2728. (Contributed by NM, 17-Jun-2017.)
Hypotheses
Ref Expression
nfreud.1  |-  F/ y
ph
nfreud.2  |-  ( ph  -> 
F/_ x A )
nfreud.3  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfrmod  |-  ( ph  ->  F/ x E* y  e.  A ps )

Proof of Theorem nfrmod
StepHypRef Expression
1 df-rmo 2564 . 2  |-  ( E* y  e.  A ps  <->  E* y ( y  e.  A  /\  ps )
)
2 nfreud.1 . . 3  |-  F/ y
ph
3 nfcvf 2454 . . . . . 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 2447 . . . 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 1775 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ( y  e.  A  /\  ps ) )
112, 10nfmod2 2169 . 2  |-  ( ph  ->  F/ x E* y
( y  e.  A  /\  ps ) )
121, 11nfxfrd 1561 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 1530   F/wnf 1534    e. wcel 1696   E*wmo 2157   F/_wnfc 2419   E*wrmo 2559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rmo 2564
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