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Theorem cbvral2vw 3226
Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 3340 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 10-Aug-2004.) Avoid ax-13 2371. (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 3171 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 𝜒))
32cbvralvw 3224 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑦𝐵 𝜒)
4 cbvral2vw.2 . . . 4 (𝑦 = 𝑤 → (𝜒𝜓))
54cbvralvw 3224 . . 3 (∀𝑦𝐵 𝜒 ↔ ∀𝑤𝐵 𝜓)
65ralbii 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 205  wral 3061
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 3062
This theorem is referenced by:  cbvral3vw  3228  fununi  6577  fiint  9271  nqereu  10870  mhmpropd  18613  efgred  19535  mplcoe5  21457  mdetunilem9  21985  fbun  23207  fbunfip  23236  caucfil  24663  pmltpc  24830  negsprop  27355  iscgrglt  27498  axcontlem10  27964  htth  29902  cdj3lem3b  31424  cdj3i  31425  dfmgc2  31905  isros  32824  rossros  32836  fipjust  41925  isotone1  42408  isotone2  42409  ntrclsiso  42427  ntrclskb  42429  ntrclsk3  42430  ntrclsk13  42431  limsuppnfd  44029  pimincfltioo  45045  incsmf  45069  decsmf  45094  mgmhmpropd  46165  catprslem  47116  isthincd2lem1  47133  isthincd2lem2  47142
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