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Theorem 19.40-2 1566
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 1565 . . 3 (∃𝑦(𝜑𝜓) → (∃𝑦𝜑 ∧ ∃𝑦𝜓))
21eximi 1534 . 2 (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(∃𝑦𝜑 ∧ ∃𝑦𝜓))
3 19.40 1565 . 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 102  wex 1424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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