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Theorem exdistr2 1807
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 1804 . 2 (∃𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑦𝑧𝜓))
21exbii 1512 1 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))
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: (None)
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