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Theorem 2exbi 41951
Description: Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
2exbi (∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓))

Proof of Theorem 2exbi
StepHypRef Expression
1 exbi 1852 . . 3 (∀𝑦(𝜑𝜓) → (∃𝑦𝜑 ↔ ∃𝑦𝜓))
21alimi 1817 . 2 (∀𝑥𝑦(𝜑𝜓) → ∀𝑥(∃𝑦𝜑 ↔ ∃𝑦𝜓))
3 exbi 1852 . 2 (∀𝑥(∃𝑦𝜑 ↔ ∃𝑦𝜓) → (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓))
42, 3syl 17 1 (∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539  wex 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815
This theorem depends on definitions:  df-bi 206  df-ex 1786
This theorem is referenced by: (None)
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