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Theorem nfrmo 2883
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 2713 . 2  |-  ( E* y  e.  A ph  <->  E* y ( y  e.  A  /\  ph )
)
2 nftru 1563 . . . 4  |-  F/ y  T.
3 nfcvf 2594 . . . . . . 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 2587 . . . . . 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 1843 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  F/ x
( y  e.  A  /\  ph ) )
109adantl 453 . . . 4  |-  ( (  T.  /\  -.  A. x  x  =  y
)  ->  F/ x
( y  e.  A  /\  ph ) )
112, 10nfmod2 2294 . . 3  |-  (  T. 
->  F/ x E* y
( y  e.  A  /\  ph ) )
1211trud 1332 . 2  |-  F/ x E* y ( y  e.  A  /\  ph )
131, 12nfxfr 1579 1  |-  F/ x E* y  e.  A ph
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    T. wtru 1325   A.wal 1549   F/wnf 1553    e. wcel 1725   E*wmo 2282   F/_wnfc 2559   E*wrmo 2708
This theorem is referenced by:  2rmorex  3138  2reurex  27935
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 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-eu 2285  df-mo 2286  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rmo 2713
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