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Theorem 19.41vv 1875
Description: Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)
Assertion
Ref Expression
19.41vv (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))
Distinct variable groups:   𝜓,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 19.41vv
StepHypRef Expression
1 19.41v 1874 . . 3 (∃𝑦(𝜑𝜓) ↔ (∃𝑦𝜑𝜓))
21exbii 1584 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(∃𝑦𝜑𝜓))
3 19.41v 1874 . 2 (∃𝑥(∃𝑦𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))
42, 3bitri 183 1 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wex 1468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  19.41vvv  1876  rabxp  4576  rexiunxp  4681  mpomptx  5862  xpassen  6724  dmaddpqlem  7197  nqpi  7198  nqnq0pi  7258  nq0nn  7262
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