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Theorem r19.41vv 3193
Description: Version of r19.41v 3191 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 3191 . . 3 (∃𝑦𝐵 (𝜑𝜓) ↔ (∃𝑦𝐵 𝜑𝜓))
21rexbii 3143 . 2 (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∃𝑥𝐴 (∃𝑦𝐵 𝜑𝜓))
3 r19.41v 3191 . 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 3015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1818  df-rex 3020
This theorem is referenced by:  genpass  9944  dfcgra2  25841  axeuclid  25963  wspthsnwspthsnon  26955  wspthsnwspthsnonOLD  26957  dya2iocnrect  30573  itg2addnclem3  33695
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