Theorem rabn0r 3420

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 3418	. 2 |- ( E. x ( x e. A /\ ph ) -> { x | ( x e. A /\ ph ) } =/= (/) )
2		df-rex 2441	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
3		df-rab 2444	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	neeq1i 2342	. 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 1472   e. wcel 2128   {cab 2143   =/= wne 2327   E.wrex 2436   {crab 2439   (/)c0 3394

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-nul 3395

This theorem is referenced by: (None)