Theorem abeq1i 2002

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 1873	. 2 |- (x e. {x | ph} <-> ph)
2		abeqri.1	. . 3 |- {x | ph} = A
3	2	eleq2i 1961	. 2 |- (x e. {x | ph} <-> x e. A)
4	1, 3	bitr3i 192	1 |- (ph <-> x e. A)

Syntax hints:   <-> wb 163   = wceq 1298   e. wcel 1300  {cab 1871

This theorem is referenced by:  bnj78 12439  bnj79OLD 12441

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

Copyright terms: Public domain