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Theorem nfrmo 2715
Description: Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
nfreu.1  |-  F/_ x A
nfreu.2  |-  F/ x ph
Assertion
Ref Expression
nfrmo  |-  F/ x E* y  e.  A ph

Proof of Theorem nfrmo
StepHypRef Expression
1 df-rmo 2551 . 2  |-  ( E* y  e.  A ph  <->  E* y ( y  e.  A  /\  ph )
)
2 nftru 1541 . . . 4  |-  F/ y  T.
3 nfcvf 2441 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
4 nfreu.1 . . . . . . . 8  |-  F/_ x A
54a1i 10 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  F/_ x A )
63, 5nfeld 2434 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  e.  A )
7 nfreu.2 . . . . . . 7  |-  F/ x ph
87a1i 10 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ x ph )
96, 8nfand 1763 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  F/ x
( y  e.  A  /\  ph ) )
109adantl 452 . . . 4  |-  ( (  T.  /\  -.  A. x  x  =  y
)  ->  F/ x
( y  e.  A  /\  ph ) )
112, 10nfmod2 2156 . . 3  |-  (  T. 
->  F/ x E* y
( y  e.  A  /\  ph ) )
1211trud 1314 . 2  |-  F/ x E* y ( y  e.  A  /\  ph )
131, 12nfxfr 1557 1  |-  F/ x E* y  e.  A ph
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    T. wtru 1307   A.wal 1527   F/wnf 1531    e. wcel 1684   E*wmo 2144   F/_wnfc 2406   E*wrmo 2546
This theorem is referenced by:  2rmorex  2969  2reurex  27959
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-mo 2148  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rmo 2551
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