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Theorem rmoun 32522
Description: "At most one" restricted existential quantifier for a union implies the same quantifier on both sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)
Assertion
Ref Expression
rmoun (∃*𝑥 ∈ (𝐴𝐵)𝜑 → (∃*𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑))

Proof of Theorem rmoun
StepHypRef Expression
1 mooran2 2554 . 2 (∃*𝑥((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)) → (∃*𝑥(𝑥𝐴𝜑) ∧ ∃*𝑥(𝑥𝐵𝜑)))
2 df-rmo 3378 . . 3 (∃*𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃*𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
3 elun 4163 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
43anbi1i 624 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
5 andir 1010 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)))
64, 5bitri 275 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)))
76mobii 2546 . . 3 (∃*𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ∃*𝑥((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)))
82, 7bitri 275 . 2 (∃*𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃*𝑥((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)))
9 df-rmo 3378 . . 3 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
10 df-rmo 3378 . . 3 (∃*𝑥𝐵 𝜑 ↔ ∃*𝑥(𝑥𝐵𝜑))
119, 10anbi12i 628 . 2 ((∃*𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑) ↔ (∃*𝑥(𝑥𝐴𝜑) ∧ ∃*𝑥(𝑥𝐵𝜑)))
121, 8, 113imtr4i 292 1 (∃*𝑥 ∈ (𝐴𝐵)𝜑 → (∃*𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  wcel 2106  ∃*wmo 2536  ∃*wrmo 3377  cun 3961
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-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-rmo 3378  df-v 3480  df-un 3968
This theorem is referenced by: (None)
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