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Theorem rmorabex 4415
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 4414 . 2  |-  ( E* x ( x  e.  A  /\  ph )  ->  { x  |  ( x  e.  A  /\  ph ) }  e.  _V )
2 df-rmo 2705 . 2  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
3 df-rab 2706 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43eleq1i 2498 . 2  |-  ( { x  e.  A  |  ph }  e.  _V  <->  { x  |  ( x  e.  A  /\  ph ) }  e.  _V )
51, 2, 43imtr4i 258 1  |-  ( E* x  e.  A ph  ->  { x  e.  A  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   E*wmo 2281   {cab 2421   E*wrmo 2700   {crab 2701   _Vcvv 2948
This theorem is referenced by:  supexd  7448
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rmo 2705  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-pr 3813
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