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Theorem 2mo2 2668
 Description: Two ways of expressing "there exists at most one ordered pair ⟨𝑥, 𝑦⟩ such that 𝜑(𝑥, 𝑦) holds. Note that this is not equivalent to ∃*𝑥∃*𝑦𝜑. See also 2mo 2669. This is the analogue of 2eu4 2675 for existential uniqueness. (Contributed by Wolf Lammen, 26-Oct-2019.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 3-Jan-2023.)
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 exdistrv 1956 . 2 (∃𝑧𝑤(∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)) ↔ (∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤)))
2 jcab 521 . . . . 5 ((𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)))
322albii 1822 . . . 4 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)))
4 19.26-2 1872 . . . 4 (∀𝑥𝑦((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)) ↔ (∀𝑥𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)))
5 19.23v 1943 . . . . . 6 (∀𝑦(𝜑𝑥 = 𝑧) ↔ (∃𝑦𝜑𝑥 = 𝑧))
65albii 1821 . . . . 5 (∀𝑥𝑦(𝜑𝑥 = 𝑧) ↔ ∀𝑥(∃𝑦𝜑𝑥 = 𝑧))
7 alcom 2160 . . . . . 6 (∀𝑥𝑦(𝜑𝑦 = 𝑤) ↔ ∀𝑦𝑥(𝜑𝑦 = 𝑤))
8 19.23v 1943 . . . . . . 7 (∀𝑥(𝜑𝑦 = 𝑤) ↔ (∃𝑥𝜑𝑦 = 𝑤))
98albii 1821 . . . . . 6 (∀𝑦𝑥(𝜑𝑦 = 𝑤) ↔ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤))
107, 9bitri 278 . . . . 5 (∀𝑥𝑦(𝜑𝑦 = 𝑤) ↔ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤))
116, 10anbi12i 629 . . . 4 ((∀𝑥𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
123, 4, 113bitri 300 . . 3 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
13122exbii 1850 . 2 (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑧𝑤(∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
14 df-mo 2557 . . 3 (∃*𝑥𝑦𝜑 ↔ ∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧))
15 df-mo 2557 . . 3 (∃*𝑦𝑥𝜑 ↔ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤))
1614, 15anbi12i 629 . 2 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ (∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤)))
171, 13, 163bitr4ri 307 1 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  ∃*wmo 2555 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-11 2158 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2557 This theorem is referenced by:  2mo  2669  2eu4  2675
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