Theorem rabeq2i 1810

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 1538	. 2 |- (x e. A <-> x e. {x e. B | ph})
3		rabid 1769	. 2 |- (x e. {x e. B | ph} <-> (x e. B /\ ph))
4	2, 3	bitr 173	1 |- (x e. A <-> (x e. B /\ ph))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  {crab 1648

This theorem is referenced by:  tfis 3127

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1652

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