Theorem rabn0r 3441

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 3439	. 2 |- ( E. x ( x e. A /\ ph ) -> { x | ( x e. A /\ ph ) } =/= (/) )
2		df-rex 2454	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
3		df-rab 2457	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	neeq1i 2355	. 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 1485   e. wcel 2141   {cab 2156   =/= wne 2340   E.wrex 2449   {crab 2452   (/)c0 3414

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-nul 3415

This theorem is referenced by: (None)