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Theorem nfrmo 3042
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 2825 . 2  |-  ( E* y  e.  A  ph  <->  E* y ( y  e.  A  /\  ph )
)
2 nftru 1609 . . . 4  |-  F/ y T.
3 nfcvf 2654 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
4 nfreu.1 . . . . . . . 8  |-  F/_ x A
54a1i 11 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  F/_ x A )
63, 5nfeld 2637 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  e.  A )
7 nfreu.2 . . . . . . 7  |-  F/ x ph
87a1i 11 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ x ph )
96, 8nfand 1872 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  F/ x
( y  e.  A  /\  ph ) )
109adantl 466 . . . 4  |-  ( ( T.  /\  -.  A. x  x  =  y
)  ->  F/ x
( y  e.  A  /\  ph ) )
112, 10nfmod2 2291 . . 3  |-  ( T. 
->  F/ x E* y
( y  e.  A  /\  ph ) )
1211trud 1388 . 2  |-  F/ x E* y ( y  e.  A  /\  ph )
131, 12nfxfr 1625 1  |-  F/ x E* y  e.  A  ph
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369   A.wal 1377   T. wtru 1380   F/wnf 1599    e. wcel 1767   E*wmo 2276   F/_wnfc 2615   E*wrmo 2820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-eu 2279  df-mo 2280  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rmo 2825
This theorem is referenced by:  2rmorex  3313  2reurex  31976
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