Theorem cbviunv 2016

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	⊢ ∪x ∈ A B = ∪y ∈ A C

Distinct variable group(s):   x,y,A   y,B   x,C

Proof of Theorem cbviunv

Step	Hyp	Ref	Expression
1		cbviunv.1	. . . . 5 ⊢ (x = y → B = C)
2	1	eleq2d 1156	. . . 4 ⊢ (x = y → (z ∈ B ↔ z ∈ C))
3	2	cbvrexv 1334	. . 3 ⊢ (∃x ∈ A z ∈ B ↔ ∃y ∈ A z ∈ C)
4	3	biabi 1181	. 2 ⊢ {z∣∃x ∈ A z ∈ B} = {z∣∃y ∈ A z ∈ C}
5		df-iun 1996	. 2 ⊢ ∪x ∈ A B = {z∣∃x ∈ A z ∈ B}
6		df-iun 1996	. 2 ⊢ ∪y ∈ A C = {z∣∃y ∈ A z ∈ C}
7	4, 5, 6	3eqtr4 1126	1 ⊢ ∪x ∈ A B = ∪y ∈ A C

Syntax hints:   → wi 2   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ∪ciun 1994

This theorem is referenced by:  trcl 3489

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-iun 1996

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