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Theorem 19.36vv 38050
Description: Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
19.36vv (∃𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦𝜑𝜓))
Distinct variable groups:   𝜓,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 19.36vv
StepHypRef Expression
1 19.36v 1906 . . 3 (∃𝑦(𝜑𝜓) ↔ (∀𝑦𝜑𝜓))
21exbii 1772 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(∀𝑦𝜑𝜓))
3 19.36v 1906 . 2 (∃𝑥(∀𝑦𝜑𝜓) ↔ (∀𝑥𝑦𝜑𝜓))
42, 3bitri 264 1 (∃𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  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  ax-5 1841  ax-6 1890
This theorem depends on definitions:  df-bi 197  df-ex 1702
This theorem is referenced by: (None)
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