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Theorem 19.40-2 1596
Description: Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
19.40-2 (∃𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 ∧ ∃𝑥𝑦𝜓))

Proof of Theorem 19.40-2
StepHypRef Expression
1 19.40 1595 . . 3 (∃𝑦(𝜑𝜓) → (∃𝑦𝜑 ∧ ∃𝑦𝜓))
21eximi 1564 . 2 (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(∃𝑦𝜑 ∧ ∃𝑦𝜓))
3 19.40 1595 . 2 (∃𝑥(∃𝑦𝜑 ∧ ∃𝑦𝜓) → (∃𝑥𝑦𝜑 ∧ ∃𝑥𝑦𝜓))
42, 3syl 14 1 (∃𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 ∧ ∃𝑥𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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