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Theorem cbviin 3956
Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
cbviun.1  |-  F/_ y B
cbviun.2  |-  F/_ x C
cbviun.3  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbviin  |-  |^|_ x  e.  A  B  =  |^|_ y  e.  A  C
Distinct variable groups:    y, A    x, A
Allowed substitution hints:    B( x, y)    C( x, y)

Proof of Theorem cbviin
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbviun.1 . . . . 5  |-  F/_ y B
21nfcri 2426 . . . 4  |-  F/ y  z  e.  B
3 cbviun.2 . . . . 5  |-  F/_ x C
43nfcri 2426 . . . 4  |-  F/ x  z  e.  C
5 cbviun.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
65eleq2d 2363 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
72, 4, 6cbvral 2773 . . 3  |-  ( A. x  e.  A  z  e.  B  <->  A. y  e.  A  z  e.  C )
87abbii 2408 . 2  |-  { z  |  A. x  e.  A  z  e.  B }  =  { z  |  A. y  e.  A  z  e.  C }
9 df-iin 3924 . 2  |-  |^|_ x  e.  A  B  =  { z  |  A. x  e.  A  z  e.  B }
10 df-iin 3924 . 2  |-  |^|_ y  e.  A  C  =  { z  |  A. y  e.  A  z  e.  C }
118, 9, 103eqtr4i 2326 1  |-  |^|_ x  e.  A  B  =  |^|_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {cab 2282   F/_wnfc 2419   A.wral 2556   |^|_ciin 3922
This theorem is referenced by:  cbviinv  3958  elrfirn2  26874
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-iin 3924
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