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Theorem r19.41vv 3120
Description: Version of r19.41v 3118 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 3118 . . 3 (∃𝑦𝐵 (𝜑𝜓) ↔ (∃𝑦𝐵 𝜑𝜓))
21rexbii 3070 . 2 (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∃𝑥𝐴 (∃𝑦𝐵 𝜑𝜓))
3 r19.41v 3118 . 2 (∃𝑥𝐴 (∃𝑦𝐵 𝜑𝜓) ↔ (∃𝑥𝐴𝑦𝐵 𝜑𝜓))
42, 3bitri 264 1 (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴𝑦𝐵 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-rex 2947
This theorem is referenced by:  genpass  9869  dfcgra2  25766  axeuclid  25888  wspthsnwspthsnon  26879  wspthsnwspthsnonOLD  26881  dya2iocnrect  30471  itg2addnclem3  33593
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