Theorem abeq1 2250

Description: Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem abeq1

Step	Hyp	Ref	Expression
1		abeq2 2249	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		eqcom 2142	. 2 |- ( { x | ph } = A <-> A = { x | ph } )
3		bicom 139	. . 3 |- ( ( ph <-> x e. A ) <-> ( x e. A <-> ph ) )
4	3	albii 1447	. 2 |- ( A. x ( ph <-> x e. A ) <-> A. x ( x e. A <-> ph ) )
5	1, 2, 4	3bitr4i 211	1 |- ( { x | ph } = A <-> A. x ( ph <-> x e. A ) )

Syntax hints:   <-> wb 104   A.wal 1330   = wceq 1332   e. wcel 1481   {cab 2126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136

This theorem is referenced by:  abbi1dv  2260  disj  3416  euabsn2  3600  dm0rn0  4764  dffo3  5575