Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cbviunf Structured version   Unicode version

Theorem cbviunf 28171
Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)
Hypotheses
Ref Expression
cbviunf.x  |-  F/_ x A
cbviunf.y  |-  F/_ y A
cbviunf.1  |-  F/_ y B
cbviunf.2  |-  F/_ x C
cbviunf.3  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbviunf  |-  U_ x  e.  A  B  =  U_ y  e.  A  C
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)

Proof of Theorem cbviunf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbviunf.x . . . 4  |-  F/_ x A
2 cbviunf.y . . . 4  |-  F/_ y A
3 cbviunf.1 . . . . 5  |-  F/_ y B
43nfcri 2573 . . . 4  |-  F/ y  z  e.  B
5 cbviunf.2 . . . . 5  |-  F/_ x C
65nfcri 2573 . . . 4  |-  F/ x  z  e.  C
7 cbviunf.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
87eleq2d 2492 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
91, 2, 4, 6, 8cbvrexf 3049 . . 3  |-  ( E. x  e.  A  z  e.  B  <->  E. y  e.  A  z  e.  C )
109abbii 2551 . 2  |-  { z  |  E. x  e.  A  z  e.  B }  =  { z  |  E. y  e.  A  z  e.  C }
11 df-iun 4301 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
12 df-iun 4301 . 2  |-  U_ y  e.  A  C  =  { z  |  E. y  e.  A  z  e.  C }
1310, 11, 123eqtr4i 2461 1  |-  U_ x  e.  A  B  =  U_ y  e.  A  C
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872   {cab 2407   F/_wnfc 2566   E.wrex 2772   U_ciun 4299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-iun 4301
This theorem is referenced by:  aciunf1lem  28266
  Copyright terms: Public domain W3C validator