Theorem rabidim1 3117

Description: Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Assertion

Ref	Expression
rabidim1	|- ( x e. { x e. A | ph } -> x e. A )

Proof of Theorem rabidim1

Step	Hyp	Ref	Expression
1		rabid 3116	. 2 |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )
2	1	simplbi 476	1 |- ( x e. { x e. A | ph } -> x e. A )

Syntax hints:   -> wi 4   e. wcel 1990   {crab 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-rab 2921

This theorem is referenced by:  frgrwopreglem5  27185  frgrwopreg  27187  ssrab2f  39300  infnsuprnmpt  39465  pimrecltpos  40919  pimrecltneg  40933  smfresal  40995  smfpimbor1lem2  41006  smflimmpt  41016  smfsupmpt  41021  smfinfmpt  41025  smflimsuplem7  41032  smflimsuplem8  41033  smflimsupmpt  41035  smfliminfmpt  41038