Theorem abeq1 2285

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 2284	. 2 |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
2		eqcom 2177	. 2 |- ( { x | ph } = A <-> A = { x | ph } )
3		bicom 140	. . 3 |- ( ( ph <-> x e. A ) <-> ( x e. A <-> ph ) )
4	3	albii 1468	. 2 |- ( A. x ( ph <-> x e. A ) <-> A. x ( x e. A <-> ph ) )
5	1, 2, 4	3bitr4i 212	1 |- ( { x | ph } = A <-> A. x ( ph <-> x e. A ) )

Syntax hints:   <-> wb 105   A.wal 1351   = wceq 1353   e. wcel 2146   {cab 2161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-11 1504  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171

This theorem is referenced by:  abbi1dv  2295  disj  3469  euabsn2  3658  dm0rn0  4837  dffo3  5655