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Theorem cbvral2vw 3373
 Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 3376 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 10-Aug-2004.) (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 3126 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 𝜒))
32cbvralvw 3361 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑦𝐵 𝜒)
4 cbvral2vw.2 . . . 4 (𝑦 = 𝑤 → (𝜒𝜓))
54cbvralvw 3361 . . 3 (∀𝑦𝐵 𝜒 ↔ ∀𝑤𝐵 𝜓)
65ralbii 3097 . 2 (∀𝑧𝐴𝑦𝐵 𝜒 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
73, 6bitri 278 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wral 3070 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 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-clel 2830  df-ral 3075 This theorem is referenced by:  cbvral3vw  3375  fununi  6410  fiint  8828  nqereu  10389  mhmpropd  18028  efgred  18941  mplcoe5  20800  mdetunilem9  21320  fbun  22540  fbunfip  22569  caucfil  23983  pmltpc  24150  iscgrglt  26407  axcontlem10  26866  htth  28800  cdj3lem3b  30322  cdj3i  30323  dfmgc2  30800  isros  31655  rossros  31667  fipjust  40637  isotone1  41124  isotone2  41125  ntrclsiso  41143  ntrclskb  41145  ntrclsk3  41146  ntrclsk13  41147  limsuppnfd  42710  pimincfltioo  43719  incsmf  43742  decsmf  43766  mgmhmpropd  44772
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