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Theorem rmoun 30260
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 2641 . 2 (∃*𝑥((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)) → (∃*𝑥(𝑥𝐴𝜑) ∧ ∃*𝑥(𝑥𝐵𝜑)))
2 df-rmo 3141 . . 3 (∃*𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃*𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
3 elun 4110 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
43anbi1i 626 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
5 andir 1006 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)))
64, 5bitri 278 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)))
76mobii 2632 . . 3 (∃*𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ∃*𝑥((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)))
82, 7bitri 278 . 2 (∃*𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃*𝑥((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)))
9 df-rmo 3141 . . 3 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
10 df-rmo 3141 . . 3 (∃*𝑥𝐵 𝜑 ↔ ∃*𝑥(𝑥𝐵𝜑))
119, 10anbi12i 629 . 2 ((∃*𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑) ↔ (∃*𝑥(𝑥𝐴𝜑) ∧ ∃*𝑥(𝑥𝐵𝜑)))
121, 8, 113imtr4i 295 1 (∃*𝑥 ∈ (𝐴𝐵)𝜑 → (∃*𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  wcel 2115  ∃*wmo 2622  ∃*wrmo 3136  cun 3917
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-mo 2624  df-clab 2803  df-cleq 2817  df-clel 2896  df-rmo 3141  df-v 3482  df-un 3924
This theorem is referenced by: (None)
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