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Theorem 19.41vvv 1784
Description: Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.)
Assertion
Ref Expression
19.41vvv (∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))
Distinct variable groups:   𝜓,𝑥   𝜓,𝑦   𝜓,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem 19.41vvv
StepHypRef Expression
1 19.41vv 1783 . . 3 (∃𝑦𝑧(𝜑𝜓) ↔ (∃𝑦𝑧𝜑𝜓))
21exbii 1496 . 2 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(∃𝑦𝑧𝜑𝜓))
3 19.41v 1782 . 2 (∃𝑥(∃𝑦𝑧𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))
42, 3bitri 173 1 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wa 97  wb 98  wex 1381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  19.41vvvv  1785  eloprabga  5591  dftpos3  5877
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