Theorem abidhb 1908

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 1001	. . 3 |- (A.y(y e. A -> A.x y e. A) -> A.yA.y(y e. A -> A.x y e. A))
2		ax-4 971	. . . 4 |- (A.x y e. A -> y e. A)
3		ax-4 971	. . . 4 |- (A.y(y e. A -> A.x y e. A) -> (y e. A -> A.x y e. A))
4	2, 3	impbid2 517	. . 3 |- (A.y(y e. A -> A.x y e. A) -> (A.x y e. A <-> y e. A))
5	1, 4	abbid 1573	. 2 |- (A.y(y e. A -> A.x y e. A) -> {y | A.x y e. A} = {y | y e. A})
6		eleq1 1531	. . . 4 |- (y = z -> (y e. A <-> z e. A))
7	6	albidv 1276	. . 3 |- (y = z -> (A.x y e. A <-> A.x z e. A))
8	7	cbvabv 1905	. 2 |- {y | A.x y e. A} = {z | A.x z e. A}
9		abid2 1577	. 2 |- {y | y e. A} = A
10	5, 8, 9	3eqtr3g 1527	1 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)

Syntax hints:   -> wi 3  A.wal 952   = wceq 954   e. wcel 956  {cab 1461

This theorem is referenced by:  hbeqd 1909  hbeld 1910  dedhb 1911  hbsbc1gd 1979  hbsbcgd 1980  hbopd 2493  hbbrd 2654  hbimad 3404  hbfvd 3721  hbfvd2 3722  hboprd 3973  hbnegd 5343

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808

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