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Theorem cbvrex2vw 3239
Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 3365 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by FL, 2-Jul-2012.) Avoid ax-13 2371. (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 3178 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 𝜒))
32cbvrexvw 3235 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑦𝐵 𝜒)
4 cbvrex2vw.2 . . . 4 (𝑦 = 𝑤 → (𝜒𝜓))
54cbvrexvw 3235 . . 3 (∃𝑦𝐵 𝜒 ↔ ∃𝑤𝐵 𝜓)
65rexbii 3094 . 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 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 1913  ax-6 1971  ax-7 2011  ax-8 2108
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-clel 2810  df-rex 3071
This theorem is referenced by:  omeu  8587  oeeui  8604  eroveu  8808  genpv  10996  bezoutlem3  16485  bezoutlem4  16486  bezout  16487  4sqlem2  16884  vdwnn  16933  efgrelexlema  19619  dyadmax  25122  2sqlem9  26937  2sq  26940  mulsval2lem  27576  precsexlemcbv  27662  legov  27874  dfcgra2  28119  pstmfval  32945  satfv0  34418  satfv0fun  34431  fmla1  34447  nn0prpwlem  35293  isbnd2  36737  nna4b4nsq  41484  oaun3lem1  42206  limsupref  44480  fourierdlem42  44944  fourierdlem54  44955  mogoldbb  46532
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