Theorem abidhb 1903

Description: Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions.

Assertion

Ref	Expression
abidhb	|- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)

Distinct variable groups:   y,z,A   x,y,z

Proof of Theorem abidhb

Step	Hyp	Ref	Expression
1		hba1 1000	. . 3 |- (A.y(y e. A -> A.x y e. A) -> A.yA.y(y e. A -> A.x y e. A))
2		ax-4 970	. . . 4 |- (A.x y e. A -> y e. A)
3		ax-4 970	. . . 4 |- (A.y(y e. A -> A.x y e. A) -> (y e. A -> A.x y e. A))
4	2, 3	impbid2 516	. . 3 |- (A.y(y e. A -> A.x y e. A) -> (A.x y e. A <-> y e. A))
5	1, 4	abbid 1568	. 2 |- (A.y(y e. A -> A.x y e. A) -> {y | A.x y e. A} = {y | y e. A})
6		eleq1 1526	. . . 4 |- (y = z -> (y e. A <-> z e. A))
7	6	albidv 1273	. . 3 |- (y = z -> (A.x y e. A <-> A.x z e. A))
8	7	cbvabv 1900	. 2 |- {y | A.x y e. A} = {z | A.x z e. A}
9		abid2 1572	. 2 |- {y | y e. A} = A
10	5, 8, 9	3eqtr3g 1522	1 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)

Syntax hints:   -> wi 3  A.wal 951   = wceq 953   e. wcel 955  {cab 1456

This theorem is referenced by:  hbeqd 1904  hbeld 1905  dedhb 1906  hbsbc1gd 1973  hbsbcgd 1974  hbopd 2488  hbbrd 2649  hbimad 3396  hbfvd 3715  hbfvd2 3716  hboprd 3967  hbnegd 5335

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

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