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Theorem 2eu3 2741
Description: Double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2392. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 23-Apr-2023.) (New usage is discouraged.)
Assertion
Ref Expression
2eu3 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Proof of Theorem 2eu3
StepHypRef Expression
1 nfmo1 2642 . . . . 5 𝑦∃*𝑦𝜑
2119.31 2238 . . . 4 (∀𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑))
32albii 1821 . . 3 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ ∀𝑥(∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑))
4 nfmo1 2642 . . . . 5 𝑥∃*𝑥𝜑
54nfal 2344 . . . 4 𝑥𝑦∃*𝑥𝜑
6519.32 2237 . . 3 (∀𝑥(∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑))
73, 6bitri 278 . 2 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑))
8 2eu1 2738 . . . . . . 7 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 ↔ (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑)))
98biimpd 232 . . . . . 6 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 → (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑)))
10 ancom 464 . . . . . 6 ((∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑))
119, 10syl6ib 254 . . . . 5 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
12 2eu1 2738 . . . . . 6 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1312biimpd 232 . . . . 5 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1411, 13jaoa 953 . . . 4 ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑦∃!𝑥𝜑 ∧ ∃!𝑥∃!𝑦𝜑) → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1514ancomsd 469 . . 3 ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
16 2exeu 2734 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
17 2exeu 2734 . . . . 5 ((∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑) → ∃!𝑦∃!𝑥𝜑)
1817ancoms 462 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑦∃!𝑥𝜑)
1916, 18jca 515 . . 3 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → (∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑))
2015, 19impbid1 228 . 2 ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
217, 20sylbi 220 1 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  wal 1536  wex 1781  ∃*wmo 2622  ∃!weu 2654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655
This theorem is referenced by: (None)
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