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Theorem 2reuswap2 3744
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 3059 . . 3 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
2 moanimv 2610 . . . 4 (∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
32albii 1813 . . 3 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
41, 3bitr4i 277 . 2 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) ↔ ∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
5 2euswapv 2621 . . 3 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃!𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃!𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑))))
6 df-reu 3375 . . . 4 (∃!𝑥𝐴𝑦𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
7 r19.42v 3188 . . . . . . 7 (∃𝑦𝐵 (𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
8 df-rex 3068 . . . . . . 7 (∃𝑦𝐵 (𝑥𝐴𝜑) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝜑)))
97, 8bitr3i 276 . . . . . 6 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝜑)))
10 an12 643 . . . . . . 7 ((𝑦𝐵 ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1110exbii 1842 . . . . . 6 (∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝜑)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
129, 11bitri 274 . . . . 5 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1312eubii 2574 . . . 4 (∃!𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
146, 13bitri 274 . . 3 (∃!𝑥𝐴𝑦𝐵 𝜑 ↔ ∃!𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
15 df-reu 3375 . . . 4 (∃!𝑦𝐵𝑥𝐴 𝜑 ↔ ∃!𝑦(𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
16 r19.42v 3188 . . . . . 6 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
17 df-rex 3068 . . . . . 6 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ ∃𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1816, 17bitr3i 276 . . . . 5 ((𝑦𝐵 ∧ ∃𝑥𝐴 𝜑) ↔ ∃𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1918eubii 2574 . . . 4 (∃!𝑦(𝑦𝐵 ∧ ∃𝑥𝐴 𝜑) ↔ ∃!𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
2015, 19bitri 274 . . 3 (∃!𝑦𝐵𝑥𝐴 𝜑 ↔ ∃!𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
215, 14, 203imtr4g 295 . 2 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))
224, 21sylbi 216 1 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1531  wex 1773  wcel 2098  ∃*wmo 2527  ∃!weu 2557  wral 3058  wrex 3067  ∃!wreu 3372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2166
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-mo 2529  df-eu 2558  df-ral 3059  df-rex 3068  df-reu 3375
This theorem is referenced by:  reuxfrd  3745
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