Theorem abeq1i 1571

Description: Equality of a class variable and a class abstraction (inference rule).

Hypothesis

Ref	Expression
abeqri.1	|- {x | ph} = A

Assertion

Ref	Expression
abeq1i	|- (ph <-> x e. A)

Proof of Theorem abeq1i

Step	Hyp	Ref	Expression
1		abid 1465	. 2 |- (x e. {x | ph} <-> ph)
2		abeqri.1	. . 3 |- {x | ph} = A
3	2	eleq2i 1538	. 2 |- (x e. {x | ph} <-> x e. A)
4	1, 3	bitr3 175	1 |- (ph <-> x e. A)

Syntax hints:   <-> wb 146   = wceq 956   e. wcel 958  {cab 1463

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

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