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Theorem cbvrex2v 3319
Description: Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
Hypotheses
Ref Expression
cbvrex2v.1 (𝑥 = 𝑧 → (𝜑𝜒))
cbvrex2v.2 (𝑦 = 𝑤 → (𝜒𝜓))
Assertion
Ref Expression
cbvrex2v (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑧,𝐴   𝑤,𝐵   𝑥,𝐵,𝑦   𝑧,𝐵,𝑦   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 cbvrex2v
StepHypRef Expression
1 cbvrex2v.1 . . . 4 (𝑥 = 𝑧 → (𝜑𝜒))
21rexbidv 3190 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 𝜒))
32cbvrexv 3311 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑦𝐵 𝜒)
4 cbvrex2v.2 . . . 4 (𝑦 = 𝑤 → (𝜒𝜓))
54cbvrexv 3311 . . 3 (∃𝑦𝐵 𝜒 ↔ ∃𝑤𝐵 𝜓)
65rexbii 3179 . 2 (∃𝑧𝐴𝑦𝐵 𝜒 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
73, 6bitri 264 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wrex 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056
This theorem is referenced by:  omeu  7836  oeeui  7853  eroveu  8011  genpv  10033  bezoutlem3  15480  bezoutlem4  15481  bezout  15482  4sqlem2  15875  vdwnn  15924  efgrelexlema  18382  dyadmax  23586  2sqlem9  25372  2sq  25375  legov  25700  dfcgra2  25941  pstmfval  30269  nn0prpwlem  32644  isbnd2  33913  fourierdlem42  40887  fourierdlem54  40898  mogoldbb  42201
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