Theorem abeq1 2188

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 2187	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		eqcom 2083	. 2 |- ( { x | ph } = A <-> A = { x | ph } )
3		bicom 138	. . 3 |- ( ( ph <-> x e. A ) <-> ( x e. A <-> ph ) )
4	3	albii 1399	. 2 |- ( A. x ( ph <-> x e. A ) <-> A. x ( x e. A <-> ph ) )
5	1, 2, 4	3bitr4i 210	1 |- ( { x | ph } = A <-> A. x ( ph <-> x e. A ) )

Syntax hints:   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   {cab 2067

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077

This theorem is referenced by:  abbi1dv  2198  disj  3292  euabsn2  3461  dm0rn0  4570  dffo3  5335