Theorem rabn0r 3523

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 3521	. 2 |- ( E. x ( x e. A /\ ph ) -> { x | ( x e. A /\ ph ) } =/= (/) )
2		df-rex 2517	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
3		df-rab 2520	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	neeq1i 2418	. 2 |- ( { x e. A | ph } =/= (/) <-> { x | ( x e. A /\ ph ) } =/= (/) )
5	1, 2, 4	3imtr4i 201	1 |- ( E. x e. A ph -> { x e. A | ph } =/= (/) )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1541   e. wcel 2202   {cab 2217   =/= wne 2403   E.wrex 2512   {crab 2515   (/)c0 3496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-nul 3497

This theorem is referenced by:  sgmnncl  15785