Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbviunvg Structured version   Visualization version   GIF version

Theorem cbviunvg 4929
 Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. Usage of this theorem is discouraged because it depends on ax-13 2379. Usage of the weaker cbviunv 4927 is preferred. (Contributed by NM, 15-Sep-2003.) (New usage is discouraged.)
Hypothesis
Ref Expression
cbviunvg.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbviunvg 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbviunvg
StepHypRef Expression
1 nfcv 2955 . 2 𝑦𝐵
2 nfcv 2955 . 2 𝑥𝐶
3 cbviunvg.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbviung 4925 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  ∪ ciun 4881 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-iun 4883 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator