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Theorem 2mo2 2533
Description: This theorem extends the idea of "at most one" to expressions in two set variables ("at most one pair 𝑥 and 𝑦". Note: this is not expressed by ∃*𝑥∃*𝑦𝜑). 2eu4 2539 relates this extension to double existential uniqueness, if at least one pair exists. (Contributed by Wolf Lammen, 26-Oct-2019.)
Assertion
Ref Expression
2mo2 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤   𝜑,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2mo2
StepHypRef Expression
1 eeanv 2164 . 2 (∃𝑧𝑤(∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)) ↔ (∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤)))
2 jcab 902 . . . . 5 ((𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)))
322albii 1736 . . . 4 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)))
4 19.26-2 1785 . . . 4 (∀𝑥𝑦((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)) ↔ (∀𝑥𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)))
5 19.23v 1887 . . . . . 6 (∀𝑦(𝜑𝑥 = 𝑧) ↔ (∃𝑦𝜑𝑥 = 𝑧))
65albii 1735 . . . . 5 (∀𝑥𝑦(𝜑𝑥 = 𝑧) ↔ ∀𝑥(∃𝑦𝜑𝑥 = 𝑧))
7 alcom 2022 . . . . . 6 (∀𝑥𝑦(𝜑𝑦 = 𝑤) ↔ ∀𝑦𝑥(𝜑𝑦 = 𝑤))
8 19.23v 1887 . . . . . . 7 (∀𝑥(𝜑𝑦 = 𝑤) ↔ (∃𝑥𝜑𝑦 = 𝑤))
98albii 1735 . . . . . 6 (∀𝑦𝑥(𝜑𝑦 = 𝑤) ↔ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤))
107, 9bitri 262 . . . . 5 (∀𝑥𝑦(𝜑𝑦 = 𝑤) ↔ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤))
116, 10anbi12i 728 . . . 4 ((∀𝑥𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
123, 4, 113bitri 284 . . 3 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
13122exbii 1763 . 2 (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑧𝑤(∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
14 mo2v 2460 . . 3 (∃*𝑥𝑦𝜑 ↔ ∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧))
15 mo2v 2460 . . 3 (∃*𝑦𝑥𝜑 ↔ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤))
1614, 15anbi12i 728 . 2 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ (∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤)))
171, 13, 163bitr4ri 291 1 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472  wex 1694  ∃*wmo 2454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031
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 2457  df-mo 2458
This theorem is referenced by:  2mo  2534  2eu4  2539
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