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

Theorem cbviinv 4496
 Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)
Hypothesis
Ref Expression
cbviunv.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbviinv 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 cbviinv
StepHypRef Expression
1 nfcv 2751 . 2 𝑦𝐵
2 nfcv 2751 . 2 𝑥𝐶
3 cbviunv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbviin 4494 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ∩ ciin 4456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-iin 4458 This theorem is referenced by:  meaiininc  39377  iinhoiicc  39565  smflimlem3  39659  smflimlem4  39660  smflimlem6  39662
 Copyright terms: Public domain W3C validator