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Theorem cbvral2vw 3239
Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 3365 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 10-Aug-2004.) Avoid ax-13 2372. (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvral2vw.1 (𝑥 = 𝑧 → (𝜑𝜒))
cbvral2vw.2 (𝑦 = 𝑤 → (𝜒𝜓))
Assertion
Ref Expression
cbvral2vw (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑤   𝑥,𝐴,𝑧   𝑥,𝑦,𝐵,𝑧   𝑤,𝐵   𝜑,𝑧   𝜓,𝑦   𝜒,𝑥   𝜒,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑤)   𝜓(𝑥,𝑧,𝑤)   𝜒(𝑦,𝑧)   𝐴(𝑦,𝑤)

Proof of Theorem cbvral2vw
StepHypRef Expression
1 cbvral2vw.1 . . . 4 (𝑥 = 𝑧 → (𝜑𝜒))
21ralbidv 3178 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 𝜒))
32cbvralvw 3235 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑦𝐵 𝜒)
4 cbvral2vw.2 . . . 4 (𝑦 = 𝑤 → (𝜒𝜓))
54cbvralvw 3235 . . 3 (∀𝑦𝐵 𝜒 ↔ ∀𝑤𝐵 𝜓)
65ralbii 3094 . 2 (∀𝑧𝐴𝑦𝐵 𝜒 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
73, 6bitri 275 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wral 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-clel 2811  df-ral 3063
This theorem is referenced by:  cbvral3vw  3241  cbvral6vw  3243  fununi  6624  fiint  9324  nqereu  10924  mhmpropd  18678  efgred  19616  mplcoe5  21595  mdetunilem9  22122  fbun  23344  fbunfip  23373  caucfil  24800  pmltpc  24967  negsprop  27509  iscgrglt  27765  axcontlem10  28231  htth  30171  cdj3lem3b  31693  cdj3i  31694  dfmgc2  32166  isros  33166  rossros  33178  fipjust  42316  isotone1  42799  isotone2  42800  ntrclsiso  42818  ntrclskb  42820  ntrclsk3  42821  ntrclsk13  42822  limsuppnfd  44418  pimincfltioo  45434  incsmf  45458  decsmf  45483  mgmhmpropd  46555  catprslem  47630  isthincd2lem1  47647  isthincd2lem2  47656
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