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Theorem 19.41vvvv 1799
Description: Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.)
Assertion
Ref Expression
19.41vvvv (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))
Distinct variable groups:   𝜓,𝑤   𝜓,𝑥   𝜓,𝑦   𝜓,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 19.41vvvv
StepHypRef Expression
1 19.41vvv 1798 . . 3 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))
21exbii 1510 . 2 (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑤(∃𝑥𝑦𝑧𝜑𝜓))
3 19.41v 1796 . 2 (∃𝑤(∃𝑥𝑦𝑧𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))
42, 3bitri 177 1 (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wex 1395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1350  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-4 1414  ax-17 1433  ax-ial 1441
This theorem depends on definitions:  df-bi 114
This theorem is referenced by: (None)
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