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Theorem rmorabex 4249
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 4248 . 2  |-  ( E* x ( x  e.  A  /\  ph )  ->  { x  |  ( x  e.  A  /\  ph ) }  e.  _V )
2 df-rmo 2564 . 2  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
3 df-rab 2565 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43eleq1i 2359 . 2  |-  ( { x  e.  A  |  ph }  e.  _V  <->  { x  |  ( x  e.  A  /\  ph ) }  e.  _V )
51, 2, 43imtr4i 257 1  |-  ( E* x  e.  A ph  ->  { x  e.  A  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   E*wmo 2157   {cab 2282   E*wrmo 2559   {crab 2560   _Vcvv 2801
This theorem is referenced by:  supexd  7220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rmo 2564  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660
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