Theorem cbviun 2586

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 1540	. . . 4 |- (x = y -> (z e. B <-> z e. C))
5	1, 2, 4	cbvrex 1797	. . 3 |- (E.x e. A z e. B <-> E.y e. A z e. C)
6	5	abbii 1574	. 2 |- {z | E.x e. A z e. B} = {z | E.y e. A z e. C}
7		df-iun 2565	. 2 |- U_x e. A B = {z | E.x e. A z e. B}
8		df-iun 2565	. 2 |- U_y e. A C = {z | E.y e. A z e. C}
9	6, 7, 8	3eqtr4 1504	1 |- U_x e. A B = U_y e. A C

Syntax hints:   -> wi 3  A.wal 953   = wceq 955   e. wcel 957  {cab 1463  E.wrex 1645  U_ciun 2563

This theorem is referenced by:  cbviunv 2587  funiunfvf 3867

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rex 1649  df-iun 2565

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