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Theorem 19.12vv 2316
Description: Special case of 19.12 2303 where its converse holds. See 19.12vvv 2037 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
19.12vv (∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem 19.12vv
StepHypRef Expression
1 19.21v 1982 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓))
21exbii 1892 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 → ∀𝑦𝜓))
3 nfv 1957 . . . 4 𝑥𝜓
43nfal 2299 . . 3 𝑥𝑦𝜓
5419.36 2216 . 2 (∃𝑥(𝜑 → ∀𝑦𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
6 19.36v 2036 . . . 4 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
76albii 1863 . . 3 (∀𝑦𝑥(𝜑𝜓) ↔ ∀𝑦(∀𝑥𝜑𝜓))
8 nfv 1957 . . . . 5 𝑦𝜑
98nfal 2299 . . . 4 𝑦𝑥𝜑
10919.21 2192 . . 3 (∀𝑦(∀𝑥𝜑𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
117, 10bitr2i 268 . 2 ((∀𝑥𝜑 → ∀𝑦𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
122, 5, 113bitri 289 1 (∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1599  wex 1823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163
This theorem depends on definitions:  df-bi 199  df-ex 1824  df-nf 1828
This theorem is referenced by: (None)
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