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Theorem 2reuswap2 3674
Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.)
Assertion
Ref Expression
2reuswap2 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2reuswap2
StepHypRef Expression
1 df-ral 3067 . . 3 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
2 moanimv 2621 . . . 4 (∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
32albii 1827 . . 3 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
41, 3bitr4i 281 . 2 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) ↔ ∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
5 2euswapv 2632 . . 3 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃!𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃!𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑))))
6 df-reu 3069 . . . 4 (∃!𝑥𝐴𝑦𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
7 r19.42v 3275 . . . . . . 7 (∃𝑦𝐵 (𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
8 df-rex 3068 . . . . . . 7 (∃𝑦𝐵 (𝑥𝐴𝜑) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝜑)))
97, 8bitr3i 280 . . . . . 6 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝜑)))
10 an12 645 . . . . . . 7 ((𝑦𝐵 ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1110exbii 1855 . . . . . 6 (∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝜑)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
129, 11bitri 278 . . . . 5 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1312eubii 2585 . . . 4 (∃!𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
146, 13bitri 278 . . 3 (∃!𝑥𝐴𝑦𝐵 𝜑 ↔ ∃!𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
15 df-reu 3069 . . . 4 (∃!𝑦𝐵𝑥𝐴 𝜑 ↔ ∃!𝑦(𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
16 r19.42v 3275 . . . . . 6 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
17 df-rex 3068 . . . . . 6 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ ∃𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1816, 17bitr3i 280 . . . . 5 ((𝑦𝐵 ∧ ∃𝑥𝐴 𝜑) ↔ ∃𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1918eubii 2585 . . . 4 (∃!𝑦(𝑦𝐵 ∧ ∃𝑥𝐴 𝜑) ↔ ∃!𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
2015, 19bitri 278 . . 3 (∃!𝑦𝐵𝑥𝐴 𝜑 ↔ ∃!𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
215, 14, 203imtr4g 299 . 2 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))
224, 21sylbi 220 1 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1541  wex 1787  wcel 2111  ∃*wmo 2538  ∃!weu 2568  wral 3062  wrex 3063  ∃!wreu 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2142  ax-11 2159  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-mo 2540  df-eu 2569  df-ral 3067  df-rex 3068  df-reu 3069
This theorem is referenced by:  reuxfrd  3675
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