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Theorem cbviin 3695
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 2172 . . . 4  |-  F/ y  z  e.  B
3 cbviun.2 . . . . 5  |-  F/_ x C
43nfcri 2172 . . . 4  |-  F/ x  z  e.  C
5 cbviun.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
65eleq2d 2107 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
72, 4, 6cbvral 2529 . . 3  |-  ( A. x  e.  A  z  e.  B  <->  A. y  e.  A  z  e.  C )
87abbii 2153 . 2  |-  { z  |  A. x  e.  A  z  e.  B }  =  { z  |  A. y  e.  A  z  e.  C }
9 df-iin 3660 . 2  |-  |^|_ x  e.  A  B  =  { z  |  A. x  e.  A  z  e.  B }
10 df-iin 3660 . 2  |-  |^|_ y  e.  A  C  =  { z  |  A. y  e.  A  z  e.  C }
118, 9, 103eqtr4i 2070 1  |-  |^|_ x  e.  A  B  =  |^|_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    e. wcel 1393   {cab 2026   F/_wnfc 2165   A.wral 2306   |^|_ciin 3658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-iin 3660
This theorem is referenced by:  cbviinv  3697
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