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Theorem 2euswapdc 2034
Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.)
Assertion
Ref Expression
2euswapdc (DECID𝑥𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑)))

Proof of Theorem 2euswapdc
StepHypRef Expression
1 excomim 1594 . . . . 5 (∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑)
21a1i 9 . . . 4 ((DECID𝑥𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → (∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑))
3 2moswapdc 2033 . . . . 5 (DECID𝑥𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑)))
43imp 122 . . . 4 ((DECID𝑥𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))
52, 4anim12d 328 . . 3 ((DECID𝑥𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ((∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑) → (∃𝑦𝑥𝜑 ∧ ∃*𝑦𝑥𝜑)))
6 eu5 1990 . . 3 (∃!𝑥𝑦𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑))
7 eu5 1990 . . 3 (∃!𝑦𝑥𝜑 ↔ (∃𝑦𝑥𝜑 ∧ ∃*𝑦𝑥𝜑))
85, 6, 73imtr4g 203 . 2 ((DECID𝑥𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑))
98ex 113 1 (DECID𝑥𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  DECID wdc 776  wal 1283  wex 1422  ∃!weu 1943  ∃*wmo 1944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947
This theorem is referenced by:  euxfr2dc  2786  2reuswapdc  2803
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