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Theorem cbvrex2vw 3238
Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 3364 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by FL, 2-Jul-2012.) Avoid ax-13 2370. (Revised by Gino Giotto, 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 3177 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 𝜒))
32cbvrexvw 3234 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑦𝐵 𝜒)
4 cbvrex2vw.2 . . . 4 (𝑦 = 𝑤 → (𝜒𝜓))
54cbvrexvw 3234 . . 3 (∃𝑦𝐵 𝜒 ↔ ∃𝑤𝐵 𝜓)
65rexbii 3093 . 2 (∃𝑧𝐴𝑦𝐵 𝜒 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
73, 6bitri 274 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wrex 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-clel 2809  df-rex 3070
This theorem is referenced by:  omeu  8588  oeeui  8605  eroveu  8809  genpv  10997  bezoutlem3  16488  bezoutlem4  16489  bezout  16490  4sqlem2  16887  vdwnn  16936  efgrelexlema  19659  dyadmax  25348  2sqlem9  27163  2sq  27166  mulsval2lem  27802  precsexlemcbv  27888  legov  28100  dfcgra2  28345  pstmfval  33171  satfv0  34644  satfv0fun  34657  fmla1  34673  nn0prpwlem  35511  isbnd2  36955  nna4b4nsq  41705  oaun3lem1  42427  limsupref  44701  fourierdlem42  45165  fourierdlem54  45176  mogoldbb  46753
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