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Theorem rmorabex 4366
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 4365 . 2  |-  ( E* x ( x  e.  A  /\  ph )  ->  { x  |  ( x  e.  A  /\  ph ) }  e.  _V )
2 df-rmo 2659 . 2  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
3 df-rab 2660 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43eleq1i 2452 . 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 1717   E*wmo 2241   {cab 2375   E*wrmo 2654   {crab 2655   _Vcvv 2901
This theorem is referenced by:  supexd  7393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-rmo 2659  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-sn 3765  df-pr 3766
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