Theorem abn0 3568

Description: Nonempty class abstraction. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Assertion

Ref	Expression
abn0	|- ({x | ph} =/= (/) <-> E.xph)

Proof of Theorem abn0

Step	Hyp	Ref	Expression
1		nfab1 2491	. . 3 |- F/_x{x | ph}
2	1	n0f 3558	. 2 |- ({x | ph} =/= (/) <-> E.x x e. {x | ph})
3		abid 2341	. . 3 |- (x e. {x | ph} <-> ph)
4	3	exbii 1582	. 2 |- (E.x x e. {x | ph} <-> E.xph)
5	2, 4	bitri 240	1 |- ({x | ph} =/= (/) <-> E.xph)

Syntax hints:   <-> wb 176  E.wex 1541   e. wcel 1710  {cab 2339   =/= wne 2516  (/)c0 3550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551

This theorem is referenced by:  ab0  3569  rabn0  3570  imasn  5018  frds  5935  mapprc  6004  map0b  6024  map0  6025