Theorem cbviunv 2587

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

Hypothesis

Ref	Expression
cbviunv.1	|- (x = y -> B = C)

Assertion

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

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

Proof of Theorem cbviunv

Step	Hyp	Ref	Expression
1		ax-17 970	. 2 |- (z e. B -> A.y z e. B)
2		ax-17 970	. 2 |- (z e. C -> A.x z e. C)
3		cbviunv.1	. 2 |- (x = y -> B = C)
4	1, 2, 3	cbviun 2586	1 |- U_x e. A B = U_y e. A C

Syntax hints:   -> wi 3   = wceq 955   e. wcel 957  U_ciun 2563

This theorem is referenced by:  iunxdif2 2595  oelim2 4219  trcl 4632

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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