MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbviinv Structured version   Unicode version

Theorem cbviinv 4124
Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)
Hypothesis
Ref Expression
cbviunv.1  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbviinv  |-  |^|_ x  e.  A  B  =  |^|_ y  e.  A  C
Distinct variable groups:    x, A    y, A    y, B    x, C
Allowed substitution hints:    B( x)    C( y)

Proof of Theorem cbviinv
StepHypRef Expression
1 nfcv 2572 . 2  |-  F/_ y B
2 nfcv 2572 . 2  |-  F/_ x C
3 cbviunv.1 . 2  |-  ( x  =  y  ->  B  =  C )
41, 2, 3cbviin 4122 1  |-  |^|_ x  e.  A  B  =  |^|_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   |^|_ciin 4087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2703  df-iin 4089
  Copyright terms: Public domain W3C validator