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Theorem 19.42vvvv 1885
 Description: Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)
Assertion
Ref Expression
19.42vvvv (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑤𝑥𝑦𝑧𝜓))
Distinct variable groups:   𝜑,𝑤   𝜑,𝑥   𝜑,𝑦   𝜑,𝑧
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 19.42vvvv
StepHypRef Expression
1 19.42vv 1883 . . 3 (∃𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑦𝑧𝜓))
212exbii 1585 . 2 (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑤𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))
3 19.42vv 1883 . 2 (∃𝑤𝑥(𝜑 ∧ ∃𝑦𝑧𝜓) ↔ (𝜑 ∧ ∃𝑤𝑥𝑦𝑧𝜓))
42, 3bitri 183 1 (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑤𝑥𝑦𝑧𝜓))
 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:  ceqsex8v  2731  enq0tr  7242
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