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Theorem cbviin 3940
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 2413 . . . 4  |-  F/ y  z  e.  B
3 cbviun.2 . . . . 5  |-  F/_ x C
43nfcri 2413 . . . 4  |-  F/ x  z  e.  C
5 cbviun.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
65eleq2d 2350 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
72, 4, 6cbvral 2760 . . 3  |-  ( A. x  e.  A  z  e.  B  <->  A. y  e.  A  z  e.  C )
87abbii 2395 . 2  |-  { z  |  A. x  e.  A  z  e.  B }  =  { z  |  A. y  e.  A  z  e.  C }
9 df-iin 3908 . 2  |-  |^|_ x  e.  A  B  =  { z  |  A. x  e.  A  z  e.  B }
10 df-iin 3908 . 2  |-  |^|_ y  e.  A  C  =  { z  |  A. y  e.  A  z  e.  C }
118, 9, 103eqtr4i 2313 1  |-  |^|_ x  e.  A  B  =  |^|_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {cab 2269   F/_wnfc 2406   A.wral 2543   |^|_ciin 3906
This theorem is referenced by:  cbviinv  3942  elrfirn2  26771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-iin 3908
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