Theorem abeq1 2460

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 2459	. 2 |- (A = {x | ph} <-> A.x(x e. A <-> ph))
2		eqcom 2355	. 2 |- ({x | ph} = A <-> A = {x | ph})
3		bicom 191	. . 3 |- ((ph <-> x e. A) <-> (x e. A <-> ph))
4	3	albii 1566	. 2 |- (A.x(ph <-> x e. A) <-> A.x(x e. A <-> ph))
5	1, 2, 4	3bitr4i 268	1 |- ({x | ph} = A <-> A.x(ph <-> x e. A))

Syntax hints:   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349

This theorem is referenced by:  abbi1dv  2470  euabsn2  3792  phidisjnn  4616  dm0rn0  4922  dffo3  5423