Theorem reqabi 2728

Description: Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)

Hypothesis

Ref	Expression
reqabi.1	|- A = { x e. B | ph }

Assertion

Ref	Expression
reqabi	|- ( x e. A <-> ( x e. B /\ ph ) )

Proof of Theorem reqabi

Step	Hyp	Ref	Expression
1		reqabi.1	. . 3 |- A = { x e. B | ph }
2	1	eleq2i 2305	. 2 |- ( x e. A <-> x e. { x e. B | ph } )
3		rabid 2727	. 2 |- ( x e. { x e. B | ph } <-> ( x e. B /\ ph ) )
4	2, 3	bitri 184	1 |- ( x e. A <-> ( x e. B /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {crab 2532

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-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-rab 2537

This theorem is referenced by:  ballotfilem2  13206