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Theorem 2mo2 2725
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 2726. This is the analogue of 2eu4 2734 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 1947 . 2 (∃𝑧𝑤(∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)) ↔ (∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤)))
2 jcab 518 . . . . 5 ((𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)))
322albii 1812 . . . 4 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)))
4 19.26-2 1863 . . . 4 (∀𝑥𝑦((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)) ↔ (∀𝑥𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)))
5 19.23v 1934 . . . . . 6 (∀𝑦(𝜑𝑥 = 𝑧) ↔ (∃𝑦𝜑𝑥 = 𝑧))
65albii 1811 . . . . 5 (∀𝑥𝑦(𝜑𝑥 = 𝑧) ↔ ∀𝑥(∃𝑦𝜑𝑥 = 𝑧))
7 alcom 2153 . . . . . 6 (∀𝑥𝑦(𝜑𝑦 = 𝑤) ↔ ∀𝑦𝑥(𝜑𝑦 = 𝑤))
8 19.23v 1934 . . . . . . 7 (∀𝑥(𝜑𝑦 = 𝑤) ↔ (∃𝑥𝜑𝑦 = 𝑤))
98albii 1811 . . . . . 6 (∀𝑦𝑥(𝜑𝑦 = 𝑤) ↔ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤))
107, 9bitri 276 . . . . 5 (∀𝑥𝑦(𝜑𝑦 = 𝑤) ↔ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤))
116, 10anbi12i 626 . . . 4 ((∀𝑥𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
123, 4, 113bitri 298 . . 3 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
13122exbii 1840 . 2 (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑧𝑤(∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
14 df-mo 2615 . . 3 (∃*𝑥𝑦𝜑 ↔ ∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧))
15 df-mo 2615 . . 3 (∃*𝑦𝑥𝜑 ↔ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤))
1614, 15anbi12i 626 . 2 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ (∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤)))
171, 13, 163bitr4ri 305 1 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526  wex 1771  ∃*wmo 2613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-11 2151
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-mo 2615
This theorem is referenced by:  2mo  2726  2eu4  2734
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