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Theorem 2eu4 2653
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 2649 for a condition under which the naive definition holds and 2exeu 2644 for a one-way implication. See 2eu5 2654 and 2eu8 2657 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 df-eu 2567 . . 3 (∃!𝑥𝑦𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑))
2 df-eu 2567 . . . 4 (∃!𝑦𝑥𝜑 ↔ (∃𝑦𝑥𝜑 ∧ ∃*𝑦𝑥𝜑))
3 excom 2160 . . . 4 (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
42, 3bianbi 627 . . 3 (∃!𝑦𝑥𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑))
51, 4anbi12i 628 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ((∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑) ∧ (∃𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑)))
6 anandi 676 . 2 ((∃𝑥𝑦𝜑 ∧ (∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑)) ↔ ((∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑) ∧ (∃𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑)))
7 2mo2 2645 . . 3 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
87anbi2i 623 . 2 ((∃𝑥𝑦𝜑 ∧ (∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑)) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
95, 6, 83bitr2i 299 1 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  wex 1776  ∃*wmo 2536  ∃!weu 2566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-11 2155
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-mo 2538  df-eu 2567
This theorem is referenced by:  2eu5  2654  2eu6  2655
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