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Theorem 2eu3 2542
Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu3 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Proof of Theorem 2eu3
StepHypRef Expression
1 nfmo1 2468 . . . . 5 𝑦∃*𝑦𝜑
2119.31 2088 . . . 4 (∀𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑))
32albii 1736 . . 3 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ ∀𝑥(∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑))
4 nfmo1 2468 . . . . 5 𝑥∃*𝑥𝜑
54nfal 2138 . . . 4 𝑥𝑦∃*𝑥𝜑
6519.32 2087 . . 3 (∀𝑥(∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑))
73, 6bitri 262 . 2 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑))
8 2eu1 2540 . . . . . . 7 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 ↔ (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑)))
98biimpd 217 . . . . . 6 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 → (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑)))
10 ancom 464 . . . . . 6 ((∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑))
119, 10syl6ib 239 . . . . 5 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1211adantld 481 . . . 4 (∀𝑦∃*𝑥𝜑 → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
13 2eu1 2540 . . . . . 6 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1413biimpd 217 . . . . 5 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1514adantrd 482 . . . 4 (∀𝑥∃*𝑦𝜑 → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1612, 15jaoi 392 . . 3 ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
17 2exeu 2536 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
18 2exeu 2536 . . . . 5 ((∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑) → ∃!𝑦∃!𝑥𝜑)
1918ancoms 467 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑦∃!𝑥𝜑)
2017, 19jca 552 . . 3 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → (∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑))
2116, 20impbid1 213 . 2 ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
227, 21sylbi 205 1 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382  wal 1472  wex 1694  ∃!weu 2457  ∃*wmo 2458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-eu 2461  df-mo 2462
This theorem is referenced by: (None)
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