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Theorem rmounid 30256
Description: A case where an "at most one" restricted existential quantifier for a union is equivalent to such a quantifier for one of the sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)
Hypothesis
Ref Expression
rmounid.1 ((𝜑𝑥𝐵) → ¬ 𝜓)
Assertion
Ref Expression
rmounid (𝜑 → (∃*𝑥 ∈ (𝐴𝐵)𝜓 ↔ ∃*𝑥𝐴 𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem rmounid
StepHypRef Expression
1 rmounid.1 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → ¬ 𝜓)
21ex 416 . . . . . . . . . . 11 (𝜑 → (𝑥𝐵 → ¬ 𝜓))
32con2d 136 . . . . . . . . . 10 (𝜑 → (𝜓 → ¬ 𝑥𝐵))
43imp 410 . . . . . . . . 9 ((𝜑𝜓) → ¬ 𝑥𝐵)
5 biorf 934 . . . . . . . . . 10 𝑥𝐵 → (𝑥𝐴 ↔ (𝑥𝐵𝑥𝐴)))
6 orcom 867 . . . . . . . . . 10 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴))
75, 6syl6bbr 292 . . . . . . . . 9 𝑥𝐵 → (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐵)))
84, 7syl 17 . . . . . . . 8 ((𝜑𝜓) → (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐵)))
9 elun 4109 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
108, 9syl6bbr 292 . . . . . . 7 ((𝜑𝜓) → (𝑥𝐴𝑥 ∈ (𝐴𝐵)))
1110pm5.32da 582 . . . . . 6 (𝜑 → ((𝜓𝑥𝐴) ↔ (𝜓𝑥 ∈ (𝐴𝐵))))
1211biancomd 467 . . . . 5 (𝜑 → ((𝜓𝑥𝐴) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝜓)))
1312bicomd 226 . . . 4 (𝜑 → ((𝑥 ∈ (𝐴𝐵) ∧ 𝜓) ↔ (𝜓𝑥𝐴)))
1413biancomd 467 . . 3 (𝜑 → ((𝑥 ∈ (𝐴𝐵) ∧ 𝜓) ↔ (𝑥𝐴𝜓)))
1514mobidv 2634 . 2 (𝜑 → (∃*𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜓) ↔ ∃*𝑥(𝑥𝐴𝜓)))
16 df-rmo 3140 . 2 (∃*𝑥 ∈ (𝐴𝐵)𝜓 ↔ ∃*𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜓))
17 df-rmo 3140 . 2 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
1815, 16, 173bitr4g 317 1 (𝜑 → (∃*𝑥 ∈ (𝐴𝐵)𝜓 ↔ ∃*𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  wcel 2115  ∃*wmo 2622  ∃*wrmo 3135  cun 3916
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 3140  df-v 3481  df-un 3923
This theorem is referenced by:  disjxun0  30323
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