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Theorem exdistr2 2034
Description: Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
Assertion
Ref Expression
exdistr2 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))
Distinct variable groups:   𝜑,𝑦   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem exdistr2
StepHypRef Expression
1 19.42vv 2032 . 2 (∃𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑦𝑧𝜓))
21exbii 1923 1 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wex 1853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854
This theorem is referenced by: (None)
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