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Theorem cbviunf 28247
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.)
Hypotheses
Ref Expression
cbviunf.x  |-  F/_ x A
cbviunf.y  |-  F/_ y A
cbviunf.1  |-  F/_ y B
cbviunf.2  |-  F/_ x C
cbviunf.3  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbviunf  |-  U_ x  e.  A  B  =  U_ y  e.  A  C
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)

Proof of Theorem cbviunf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbviunf.x . . . 4  |-  F/_ x A
2 cbviunf.y . . . 4  |-  F/_ y A
3 cbviunf.1 . . . . 5  |-  F/_ y B
43nfcri 2606 . . . 4  |-  F/ y  z  e.  B
5 cbviunf.2 . . . . 5  |-  F/_ x C
65nfcri 2606 . . . 4  |-  F/ x  z  e.  C
7 cbviunf.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
87eleq2d 2534 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
91, 2, 4, 6, 8cbvrexf 3000 . . 3  |-  ( E. x  e.  A  z  e.  B  <->  E. y  e.  A  z  e.  C )
109abbii 2587 . 2  |-  { z  |  E. x  e.  A  z  e.  B }  =  { z  |  E. y  e.  A  z  e.  C }
11 df-iun 4271 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
12 df-iun 4271 . 2  |-  U_ y  e.  A  C  =  { z  |  E. y  e.  A  z  e.  C }
1310, 11, 123eqtr4i 2503 1  |-  U_ x  e.  A  B  =  U_ y  e.  A  C
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   {cab 2457   F/_wnfc 2599   E.wrex 2757   U_ciun 4269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-iun 4271
This theorem is referenced by:  aciunf1lem  28339
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