Theorem rabn0r 3384

Description: Nonempty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)

Assertion

Ref	Expression
rabn0r	|- ( E. x e. A ph -> { x e. A | ph } =/= (/) )

Proof of Theorem rabn0r

Step	Hyp	Ref	Expression
1		abn0r 3382	. 2 |- ( E. x ( x e. A /\ ph ) -> { x | ( x e. A /\ ph ) } =/= (/) )
2		df-rex 2420	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
3		df-rab 2423	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	neeq1i 2321	. 2 |- ( { x e. A | ph } =/= (/) <-> { x | ( x e. A /\ ph ) } =/= (/) )
5	1, 2, 4	3imtr4i 200	1 |- ( E. x e. A ph -> { x e. A | ph } =/= (/) )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1468   e. wcel 1480   {cab 2123   =/= wne 2306   E.wrex 2415   {crab 2418   (/)c0 3358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-nul 3359

This theorem is referenced by: (None)