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Theorem 19.37vv 37409
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 1896 . . 3 (∃𝑦(𝜓𝜑) ↔ (𝜓 → ∃𝑦𝜑))
21exbii 1763 . 2 (∃𝑥𝑦(𝜓𝜑) ↔ ∃𝑥(𝜓 → ∃𝑦𝜑))
3 19.37v 1896 . 2 (∃𝑥(𝜓 → ∃𝑦𝜑) ↔ (𝜓 → ∃𝑥𝑦𝜑))
42, 3bitri 262 1 (∃𝑥𝑦(𝜓𝜑) ↔ (𝜓 → ∃𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by: (None)
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