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Theorem cbviun 4003
 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
cbviun.1
cbviun.2
cbviun.3
Assertion
Ref Expression
cbviun
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   (,)

Proof of Theorem cbviun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 cbviun.1 . . . . 5
21nfcri 2483 . . . 4
3 cbviun.2 . . . . 5
43nfcri 2483 . . . 4
5 cbviun.3 . . . . 5
65eleq2d 2420 . . . 4
72, 4, 6cbvrex 2832 . . 3
87abbii 2465 . 2
9 df-iun 3971 . 2
10 df-iun 3971 . 2
118, 9, 103eqtr4i 2383 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1642   wcel 1710  cab 2339  wnfc 2476  wrex 2615  ciun 3969 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-iun 3971 This theorem is referenced by:  cbviunv  4005  funiunfvf  5468  fmpt2x  5730
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