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Theorem 19.12vv 2341
Description: Special case of 19.12 2325 where its converse holds. See 19.12vvv 2074 for a version with a dv 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 2019 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓))
21exbii 1923 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 → ∀𝑦𝜓))
3 nfv 1994 . . . 4 𝑥𝜓
43nfal 2316 . . 3 𝑥𝑦𝜓
5419.36 2253 . 2 (∃𝑥(𝜑 → ∀𝑦𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
6 19.36v 2071 . . . 4 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
76albii 1894 . . 3 (∀𝑦𝑥(𝜑𝜓) ↔ ∀𝑦(∀𝑥𝜑𝜓))
8 nfv 1994 . . . . 5 𝑦𝜑
98nfal 2316 . . . 4 𝑦𝑥𝜑
10919.21 2230 . . 3 (∀𝑦(∀𝑥𝜑𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
117, 10bitr2i 265 . 2 ((∀𝑥𝜑 → ∀𝑦𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
122, 5, 113bitri 286 1 (∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1628  wex 1851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-11 2189  ax-12 2202
This theorem depends on definitions:  df-bi 197  df-ex 1852  df-nf 1857
This theorem is referenced by: (None)
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