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Theorem 19.41vvv 1800
 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 1799 . . 3 (∃𝑦𝑧(𝜑𝜓) ↔ (∃𝑦𝑧𝜑𝜓))
21exbii 1512 . 2 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(∃𝑦𝑧𝜑𝜓))
3 19.41v 1798 . 2 (∃𝑥(∃𝑦𝑧𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))
42, 3bitri 177 1 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   ∧ wa 101   ↔ wb 102  ∃wex 1397 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 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443 This theorem depends on definitions:  df-bi 114 This theorem is referenced by:  19.41vvvv  1801  eloprabga  5619  dftpos3  5908
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