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Theorem 19.40-2 1811
Description: Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with two 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 1794 . . 3 (∃𝑦(𝜑𝜓) → (∃𝑦𝜑 ∧ ∃𝑦𝜓))
21eximi 1759 . 2 (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(∃𝑦𝜑 ∧ ∃𝑦𝜓))
3 19.40 1794 . 2 (∃𝑥(∃𝑦𝜑 ∧ ∃𝑦𝜓) → (∃𝑥𝑦𝜑 ∧ ∃𝑥𝑦𝜓))
42, 3syl 17 1 (∃𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 ∧ ∃𝑥𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
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