Theorem rabeq2i 2291

Description: Inference rule from equality of a class variable and a restricted class abstraction.

Hypothesis

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

Assertion

Ref	Expression
rabeq2i	|- (x e. A <-> (x e. B /\ ph))

Proof of Theorem rabeq2i

Step	Hyp	Ref	Expression
1		rabeqi.1	. . 3 |- A = {x e. B | ph}
2	1	eleq2i 1961	. 2 |- (x e. A <-> x e. {x e. B | ph})
3		rabid 2253	. 2 |- (x e. {x e. B | ph} <-> (x e. B /\ ph))
4	2, 3	bitri 190	1 |- (x e. A <-> (x e. B /\ ph))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {crab 2108

This theorem is referenced by:  tfis 3938  bnj1220 12994  bnj1471 13150  bnj1476 13156  bnj1526 13178  bnj1533 13182  bnj1538 13186  bnj70 13205  bnj1152 13437  bnj1284 13482  bnj1388 13514

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rab 2112

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