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Theorem 19.42vvvv 1925
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 1923 . . 3 (∃𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑦𝑧𝜓))
212exbii 1617 . 2 (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑤𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))
3 19.42vv 1923 . 2 (∃𝑤𝑥(𝜑 ∧ ∃𝑦𝑧𝜓) ↔ (𝜑 ∧ ∃𝑤𝑥𝑦𝑧𝜓))
42, 3bitri 184 1 (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑤𝑥𝑦𝑧𝜓))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wex 1503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  ceqsex8v  2797  enq0tr  7452
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