Theorem abeq1 1545

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 1544	. 2 |- (A = {x | ph} <-> A.x(x e. A <-> ph))
2		eqcom 1453	. 2 |- ({x | ph} = A <-> A = {x | ph})
3		bicom 518	. . 3 |- ((ph <-> x e. A) <-> (x e. A <-> ph))
4	3	albii 975	. 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 950   = wceq 1099   e. wcel 1105  {cab 1440

This theorem is referenced by:  abbi1dv 1555  disj 2282  eusn 2416  dm0rn0 3287  dffo3 3758

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449

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