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Theorem 19.12b 31831
Description: Version of 19.12vv 2216 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
19.12b.1 𝑦𝜑
19.12b.2 𝑥𝜓
Assertion
Ref Expression
19.12b (∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem 19.12b
StepHypRef Expression
1 19.12b.1 . . . 4 𝑦𝜑
2119.21 2113 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓))
32exbii 1814 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 → ∀𝑦𝜓))
4 19.12b.2 . . . 4 𝑥𝜓
54nfal 2191 . . 3 𝑥𝑦𝜓
6519.36 2136 . 2 (∃𝑥(𝜑 → ∀𝑦𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
7419.36 2136 . . . 4 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
87albii 1787 . . 3 (∀𝑦𝑥(𝜑𝜓) ↔ ∀𝑦(∀𝑥𝜑𝜓))
91nfal 2191 . . . 4 𝑦𝑥𝜑
10919.21 2113 . . 3 (∀𝑦(∀𝑥𝜑𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
118, 10bitr2i 265 . 2 ((∀𝑥𝜑 → ∀𝑦𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
123, 6, 113bitri 286 1 (∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wex 1744  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-ex 1745  df-nf 1750
This theorem is referenced by: (None)
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