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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 yB
cbviun.2 xC
cbviun.3 (x = yB = C)
Assertion
Ref Expression
cbviun x A B = y A C
Distinct variable groups:   y,A   x,A
Allowed substitution hints:   B(x,y)   C(x,y)

Proof of Theorem cbviun
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 cbviun.1 . . . . 5 yB
21nfcri 2483 . . . 4 y z B
3 cbviun.2 . . . . 5 xC
43nfcri 2483 . . . 4 x z C
5 cbviun.3 . . . . 5 (x = yB = C)
65eleq2d 2420 . . . 4 (x = y → (z Bz C))
72, 4, 6cbvrex 2832 . . 3 (x A z By A z C)
87abbii 2465 . 2 {z x A z B} = {z y A z C}
9 df-iun 3971 . 2 x A B = {z x A z B}
10 df-iun 3971 . 2 y A C = {z y A z C}
118, 9, 103eqtr4i 2383 1 x A B = y A C
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 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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