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Theorem cbviin 4307
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 2606 . . . 4  |-  F/ y  z  e.  B
3 cbviun.2 . . . . 5  |-  F/_ x C
43nfcri 2606 . . . 4  |-  F/ x  z  e.  C
5 cbviun.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
65eleq2d 2534 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
72, 4, 6cbvral 3001 . . 3  |-  ( A. x  e.  A  z  e.  B  <->  A. y  e.  A  z  e.  C )
87abbii 2587 . 2  |-  { z  |  A. x  e.  A  z  e.  B }  =  { z  |  A. y  e.  A  z  e.  C }
9 df-iin 4272 . 2  |-  |^|_ x  e.  A  B  =  { z  |  A. x  e.  A  z  e.  B }
10 df-iin 4272 . 2  |-  |^|_ y  e.  A  C  =  { z  |  A. y  e.  A  z  e.  C }
118, 9, 103eqtr4i 2503 1  |-  |^|_ x  e.  A  B  =  |^|_ y  e.  A  C
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   {cab 2457   F/_wnfc 2599   A.wral 2756   |^|_ciin 4270
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-iin 4272
This theorem is referenced by:  cbviinv  4309  elrfirn2  35609
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