Theorem rabn0r 3394

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 3392	. 2 |- ( E. x ( x e. A /\ ph ) -> { x | ( x e. A /\ ph ) } =/= (/) )
2		df-rex 2423	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
3		df-rab 2426	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	neeq1i 2324	. 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 1469   e. wcel 1481   {cab 2126   =/= wne 2309   E.wrex 2418   {crab 2421   (/)c0 3368

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-nul 3369

This theorem is referenced by: (None)