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Theorem rmorabex 4651
Description: Restricted "at most one" existence implies a restricted class abstraction exists. (Contributed by NM, 17-Jun-2017.)
Assertion
Ref Expression
rmorabex  |-  ( E* x  e.  A  ph  ->  { x  e.  A  |  ph }  e.  _V )

Proof of Theorem rmorabex
StepHypRef Expression
1 moabex 4650 . 2  |-  ( E* x ( x  e.  A  /\  ph )  ->  { x  |  ( x  e.  A  /\  ph ) }  e.  _V )
2 df-rmo 2762 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
3 df-rab 2763 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43eleq1i 2479 . 2  |-  ( { x  e.  A  |  ph }  e.  _V  <->  { x  |  ( x  e.  A  /\  ph ) }  e.  _V )
51, 2, 43imtr4i 266 1  |-  ( E* x  e.  A  ph  ->  { x  e.  A  |  ph }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1842   E*wmo 2239   {cab 2387   E*wrmo 2757   {crab 2758   _Vcvv 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rmo 2762  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-sn 3973  df-pr 3975
This theorem is referenced by:  supexd  7946
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