MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvrex2vw Structured version   Visualization version   GIF version

Theorem cbvrex2vw 3241
Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 3368 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by FL, 2-Jul-2012.) Avoid ax-13 2376. (Revised by GG, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvrex2vw.1 (𝑥 = 𝑧 → (𝜑𝜒))
cbvrex2vw.2 (𝑦 = 𝑤 → (𝜒𝜓))
Assertion
Ref Expression
cbvrex2vw (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑤   𝑥,𝐴,𝑧   𝑤,𝐵   𝑥,𝐵,𝑦,𝑧   𝜒,𝑤   𝜒,𝑥   𝜑,𝑧   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑤)   𝜓(𝑥,𝑧,𝑤)   𝜒(𝑦,𝑧)   𝐴(𝑦,𝑤)

Proof of Theorem cbvrex2vw
StepHypRef Expression
1 cbvrex2vw.1 . . . 4 (𝑥 = 𝑧 → (𝜑𝜒))
21rexbidv 3178 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 𝜒))
32cbvrexvw 3237 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑦𝐵 𝜒)
4 cbvrex2vw.2 . . . 4 (𝑦 = 𝑤 → (𝜒𝜓))
54cbvrexvw 3237 . . 3 (∃𝑦𝐵 𝜒 ↔ ∃𝑤𝐵 𝜓)
65rexbii 3093 . 2 (∃𝑧𝐴𝑦𝐵 𝜒 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
73, 6bitri 275 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wrex 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-clel 2815  df-rex 3070
This theorem is referenced by:  omeu  8624  oeeui  8641  eroveu  8853  genpv  11040  bezoutlem3  16579  bezoutlem4  16580  bezout  16581  4sqlem2  16988  vdwnn  17037  efgrelexlema  19768  dyadmax  25634  2sqlem9  27472  2sq  27475  mulsval2lem  28137  mulsunif2  28197  precsexlemcbv  28231  legov  28594  dfcgra2  28839  gsumwun  33069  pstmfval  33896  satfv0  35364  satfv0fun  35377  fmla1  35393  nn0prpwlem  36324  isbnd2  37791  hashnexinjle  42131  aks6d1c6lem3  42174  nna4b4nsq  42675  oaun3lem1  43392  limsupref  45705  fourierdlem42  46169  fourierdlem54  46180  mogoldbb  47777
  Copyright terms: Public domain W3C validator