Theorem abeq1 1569

Description: Equality of a class variable and a class abstraction.

Assertion

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

Distinct variable group:   x,A

Proof of Theorem abeq1

Step	Hyp	Ref	Expression
1		abeq2 1568	. 2 |- (A = {x | ph} <-> A.x(x e. A <-> ph))
2		eqcom 1477	. 2 |- ({x | ph} = A <-> A = {x | ph})
3		bicom 520	. . 3 |- ((ph <-> x e. A) <-> (x e. A <-> ph))
4	3	albii 999	. 2 |- (A.x(ph <-> x e. A) <-> A.x(x e. A <-> ph))
5	1, 2, 4	3bitr4 183	1 |- ({x | ph} = A <-> A.x(ph <-> x e. A))

Syntax hints:   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  {cab 1463

This theorem is referenced by:  abbi1dv 1579  disj 2311  eusn 2446  dm0rn0 3330  dffo3 3819  homcard 10539

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472

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