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Theorem 2eu4 2543
Description: This theorem provides us with a definition of double existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"). Naively one might think (incorrectly) that it could be defined by ∃!𝑥∃!𝑦𝜑. See 2eu1 2540 for a condition under which the naive definition holds and 2exeu 2536 for a one-way implication. See 2eu5 2544 and 2eu8 2547 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.)
Assertion
Ref Expression
2eu4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤   𝜑,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2eu4
StepHypRef Expression
1 eu5 2483 . . 3 (∃!𝑥𝑦𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑))
2 eu5 2483 . . . 4 (∃!𝑦𝑥𝜑 ↔ (∃𝑦𝑥𝜑 ∧ ∃*𝑦𝑥𝜑))
3 excom 2027 . . . . 5 (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
43anbi1i 726 . . . 4 ((∃𝑦𝑥𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑))
52, 4bitri 262 . . 3 (∃!𝑦𝑥𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑))
61, 5anbi12i 728 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ((∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑) ∧ (∃𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑)))
7 anandi 866 . 2 ((∃𝑥𝑦𝜑 ∧ (∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑)) ↔ ((∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑) ∧ (∃𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑)))
8 2mo2 2537 . . 3 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
98anbi2i 725 . 2 ((∃𝑥𝑦𝜑 ∧ (∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑)) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
106, 7, 93bitr2i 286 1 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  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 2032
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:  2eu5  2544  2eu6  2545
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