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Theorem rmorabex 4707
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 4706 . 2  |-  ( E* x ( x  e.  A  /\  ph )  ->  { x  |  ( x  e.  A  /\  ph ) }  e.  _V )
2 df-rmo 2822 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
3 df-rab 2823 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43eleq1i 2544 . 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 369    e. wcel 1767   E*wmo 2276   {cab 2452   E*wrmo 2817   {crab 2818   _Vcvv 3113
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rmo 2822  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-pr 4030
This theorem is referenced by:  supexd  7909
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