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Metamath Proof Explorer

Theorem cbviun

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006) (Revised by Andrew Salmon, 25-Jul-2011) Add disjoint variable condition to avoid ax-13 . See cbviung for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024)

Ref Expression
Hypotheses cbviun.1 
|- F/_ y B
cbviun.2 
|- F/_ x C
cbviun.3 
|- ( x = y -> B = C )
Assertion cbviun 
|- U_ x e. A B = U_ y e. A C

Proof

Step Hyp Ref Expression
1 cbviun.1 
 |-  F/_ y B
2 cbviun.2 
 |-  F/_ x C
3 cbviun.3 
 |-  ( x = y -> B = C )
4 1 nfcri 
 |-  F/ y z e. B
5 2 nfcri 
 |-  F/ x z e. C
6 3 eleq2d 
 |-  ( x = y -> ( z e. B <-> z e. C ) )
7 4 5 6 cbvrexw 
 |-  ( E. x e. A z e. B <-> E. y e. A z e. C )
8 7 abbii 
 |-  { z | E. x e. A z e. B } = { z | E. y e. A z e. C }
9 df-iun 
 |-  U_ x e. A B = { z | E. x e. A z e. B }
10 df-iun 
 |-  U_ y e. A C = { z | E. y e. A z e. C }
11 8 9 10 3eqtr4i 
 |-  U_ x e. A B = U_ y e. A C