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Theorem 19.37vv 37606
 Description: Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
19.37vv (∃𝑥𝑦(𝜓𝜑) ↔ (𝜓 → ∃𝑥𝑦𝜑))
Distinct variable groups:   𝜓,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 19.37vv
StepHypRef Expression
1 19.37v 1897 . . 3 (∃𝑦(𝜓𝜑) ↔ (𝜓 → ∃𝑦𝜑))
21exbii 1764 . 2 (∃𝑥𝑦(𝜓𝜑) ↔ ∃𝑥(𝜓 → ∃𝑦𝜑))
3 19.37v 1897 . 2 (∃𝑥(𝜓 → ∃𝑦𝜑) ↔ (𝜓 → ∃𝑥𝑦𝜑))
42, 3bitri 263 1 (∃𝑥𝑦(𝜓𝜑) ↔ (𝜓 → ∃𝑥𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by: (None)
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