Theorem rmorabex 4928

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

Step	Hyp	Ref	Expression
1		moabex 4927	. 2 |- ( E* x ( x e. A /\ ph ) -> { x | ( x e. A /\ ph ) } e. _V )
2		df-rmo 2920	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
3		df-rab 2921	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	eleq1i 2692	. 2 |- ( { x e. A | ph } e. _V <-> { x | ( x e. A /\ ph ) } e. _V )
5	1, 2, 4	3imtr4i 281	1 |- ( E* x e. A ph -> { x e. A | ph } e. _V )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   E*wmo 2471   {cab 2608   E*wrmo 2915   {crab 2916   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rmo 2920  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by:  supexd  8359