Theorem cbviun 2589

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis.

Hypotheses

Ref	Expression
cbviun.1	|- (z e. B -> A.y z e. B)
cbviun.2	|- (z e. C -> A.x z e. C)
cbviun.3	|- (x = y -> B = C)

Assertion

Ref	Expression
cbviun	|- U_x e. A B = U_y e. A C

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

Proof of Theorem cbviun

Step	Hyp	Ref	Expression
1		cbviun.1	. . . 4 |- (z e. B -> A.y z e. B)
2		cbviun.2	. . . 4 |- (z e. C -> A.x z e. C)
3		cbviun.3	. . . . 5 |- (x = y -> B = C)
4	3	eleq2d 1541	. . . 4 |- (x = y -> (z e. B <-> z e. C))
5	1, 2, 4	cbvrex 1799	. . 3 |- (E.x e. A z e. B <-> E.y e. A z e. C)
6	5	abbii 1575	. 2 |- {z | E.x e. A z e. B} = {z | E.y e. A z e. C}
7		df-iun 2568	. 2 |- U_x e. A B = {z | E.x e. A z e. B}
8		df-iun 2568	. 2 |- U_y e. A C = {z | E.y e. A z e. C}
9	6, 7, 8	3eqtr4 1505	1 |- U_x e. A B = U_y e. A C

Syntax hints:   -> wi 3  A.wal 954   = wceq 956   e. wcel 958  {cab 1463  E.wrex 1646  U_ciun 2566

This theorem is referenced by:  cbviunv 2590  funiunfvf 3870

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  df-rex 1650  df-iun 2568

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