Theorem abn0r 3432

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

Assertion

Ref	Expression
abn0r	|- ( E. x ph -> { x | ph } =/= (/) )

Proof of Theorem abn0r

Step	Hyp	Ref	Expression
1		abid 2153	. . 3 |- ( x e. { x | ph } <-> ph )
2	1	exbii 1593	. 2 |- ( E. x x e. { x | ph } <-> E. x ph )
3		nfab1 2309	. . 3 |- F/_ x { x | ph }
4	3	n0rf 3420	. 2 |- ( E. x x e. { x | ph } -> { x | ph } =/= (/) )
5	2, 4	sylbir 134	1 |- ( E. x ph -> { x | ph } =/= (/) )

Syntax hints:   -> wi 4   E.wex 1480   e. wcel 2136   {cab 2151   =/= wne 2335   (/)c0 3408

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-v 2727  df-dif 3117  df-nul 3409

This theorem is referenced by:  rabn0r  3434