Theorem abeq1 2280

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 2279	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		eqcom 2172	. 2 |- ( { x | ph } = A <-> A = { x | ph } )
3		bicom 139	. . 3 |- ( ( ph <-> x e. A ) <-> ( x e. A <-> ph ) )
4	3	albii 1463	. 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 1346   = wceq 1348   e. wcel 2141   {cab 2156

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166

This theorem is referenced by:  abbi1dv  2290  disj  3463  euabsn2  3652  dm0rn0  4828  dffo3  5643