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Theorem r19.41vv 3227
Description: Version of r19.41v 3189 with two quantifiers. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Assertion
Ref Expression
r19.41vv (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴𝑦𝐵 𝜑𝜓))
Distinct variable groups:   𝜓,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem r19.41vv
StepHypRef Expression
1 r19.41v 3189 . . 3 (∃𝑦𝐵 (𝜑𝜓) ↔ (∃𝑦𝐵 𝜑𝜓))
21rexbii 3094 . 2 (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∃𝑥𝐴 (∃𝑦𝐵 𝜑𝜓))
3 r19.41v 3189 . 2 (∃𝑥𝐴 (∃𝑦𝐵 𝜑𝜓) ↔ (∃𝑥𝐴𝑦𝐵 𝜑𝜓))
42, 3bitri 275 1 (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴𝑦𝐵 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-rex 3071
This theorem is referenced by:  genpass  11049  mulsuniflem  28175  addsdilem2  28178  mulsasslem1  28189  mulsasslem2  28190  dfcgra2  28838  axeuclid  28978  wspthsnwspthsnon  29936  dya2iocnrect  34283  satfv0  35363  satfv1  35368  satf0  35377  itg2addnclem3  37680  prprelprb  47504  prprspr2  47505
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