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Theorem 2mo2 2548
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 2554 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 2180 . 2 (∃𝑧𝑤(∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)) ↔ (∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤)))
2 jcab 906 . . . . 5 ((𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)))
322albii 1746 . . . 4 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)))
4 19.26-2 1797 . . . 4 (∀𝑥𝑦((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)) ↔ (∀𝑥𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)))
5 19.23v 1900 . . . . . 6 (∀𝑦(𝜑𝑥 = 𝑧) ↔ (∃𝑦𝜑𝑥 = 𝑧))
65albii 1745 . . . . 5 (∀𝑥𝑦(𝜑𝑥 = 𝑧) ↔ ∀𝑥(∃𝑦𝜑𝑥 = 𝑧))
7 alcom 2035 . . . . . 6 (∀𝑥𝑦(𝜑𝑦 = 𝑤) ↔ ∀𝑦𝑥(𝜑𝑦 = 𝑤))
8 19.23v 1900 . . . . . . 7 (∀𝑥(𝜑𝑦 = 𝑤) ↔ (∃𝑥𝜑𝑦 = 𝑤))
98albii 1745 . . . . . 6 (∀𝑦𝑥(𝜑𝑦 = 𝑤) ↔ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤))
107, 9bitri 264 . . . . 5 (∀𝑥𝑦(𝜑𝑦 = 𝑤) ↔ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤))
116, 10anbi12i 732 . . . 4 ((∀𝑥𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
123, 4, 113bitri 286 . . 3 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
13122exbii 1773 . 2 (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑧𝑤(∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
14 mo2v 2475 . . 3 (∃*𝑥𝑦𝜑 ↔ ∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧))
15 mo2v 2475 . . 3 (∃*𝑦𝑥𝜑 ↔ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤))
1614, 15anbi12i 732 . 2 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ (∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤)))
171, 13, 163bitr4ri 293 1 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1479  wex 1702  ∃*wmo 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1703  df-nf 1708  df-eu 2472  df-mo 2473
This theorem is referenced by:  2mo  2549  2eu4  2554
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