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Theorem 19.12vvv 2087
Description: Version of 19.12vv 2344 with a disjoint variable condition, requiring fewer axioms. See also 19.12 2330. (Contributed by BJ, 18-Mar-2020.)
Assertion
Ref Expression
19.12vvv (∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem 19.12vvv
StepHypRef Expression
1 19.21v 2034 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓))
21exbii 1943 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 → ∀𝑦𝜓))
3 19.36v 2085 . 2 (∃𝑥(𝜑 → ∀𝑦𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
4 19.36v 2085 . . . 4 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
54albii 1914 . . 3 (∀𝑦𝑥(𝜑𝜓) ↔ ∀𝑦(∀𝑥𝜑𝜓))
6 19.21v 2034 . . 3 (∀𝑦(∀𝑥𝜑𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
75, 6bitr2i 267 . 2 ((∀𝑥𝜑 → ∀𝑦𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
82, 3, 73bitri 288 1 (∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1650  wex 1874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070
This theorem depends on definitions:  df-bi 198  df-ex 1875
This theorem is referenced by: (None)
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