Theorem abeq1 2000

Description: Equality of a class variable and a class abstraction.

Assertion

Ref	Expression
abeq1	|- ({x | ph} = A <-> A.x(ph <-> x e. A))

Distinct variable group:   x,A

Proof of Theorem abeq1

Step	Hyp	Ref	Expression
1		abeq2 1999	. 2 |- (A = {x | ph} <-> A.x(x e. A <-> ph))
2		eqcom 1886	. 2 |- ({x | ph} = A <-> A = {x | ph})
3		bicom 579	. . 3 |- ((ph <-> x e. A) <-> (x e. A <-> ph))
4	3	albii 1346	. 2 |- (A.x(ph <-> x e. A) <-> A.x(x e. A <-> ph))
5	1, 2, 4	3bitr4i 200	1 |- ({x | ph} = A <-> A.x(ph <-> x e. A))

Syntax hints:   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871

This theorem is referenced by:  abbi1dv 2010  disj 2914  euabsn 3095  dfepfr 3640  dm0rn0 4175  dffo3 4792  dfon3 14072  inpc 14619  grpdivfo 14737  homcard 14893  domleqt 15020  inficlALT 15372

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  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

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