Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rmounid Structured version   Visualization version   GIF version

Theorem rmounid 30852
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 413 . . . . . . . . . . 11 (𝜑 → (𝑥𝐵 → ¬ 𝜓))
32con2d 134 . . . . . . . . . 10 (𝜑 → (𝜓 → ¬ 𝑥𝐵))
43imp 407 . . . . . . . . 9 ((𝜑𝜓) → ¬ 𝑥𝐵)
5 biorf 934 . . . . . . . . . 10 𝑥𝐵 → (𝑥𝐴 ↔ (𝑥𝐵𝑥𝐴)))
6 orcom 867 . . . . . . . . . 10 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴))
75, 6bitr4di 289 . . . . . . . . 9 𝑥𝐵 → (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐵)))
84, 7syl 17 . . . . . . . 8 ((𝜑𝜓) → (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐵)))
9 elun 4088 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
108, 9bitr4di 289 . . . . . . 7 ((𝜑𝜓) → (𝑥𝐴𝑥 ∈ (𝐴𝐵)))
1110pm5.32da 579 . . . . . 6 (𝜑 → ((𝜓𝑥𝐴) ↔ (𝜓𝑥 ∈ (𝐴𝐵))))
1211biancomd 464 . . . . 5 (𝜑 → ((𝜓𝑥𝐴) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝜓)))
1312bicomd 222 . . . 4 (𝜑 → ((𝑥 ∈ (𝐴𝐵) ∧ 𝜓) ↔ (𝜓𝑥𝐴)))
1413biancomd 464 . . 3 (𝜑 → ((𝑥 ∈ (𝐴𝐵) ∧ 𝜓) ↔ (𝑥𝐴𝜓)))
1514mobidv 2551 . 2 (𝜑 → (∃*𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜓) ↔ ∃*𝑥(𝑥𝐴𝜓)))
16 df-rmo 3074 . 2 (∃*𝑥 ∈ (𝐴𝐵)𝜓 ↔ ∃*𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜓))
17 df-rmo 3074 . 2 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
1815, 16, 173bitr4g 314 1 (𝜑 → (∃*𝑥 ∈ (𝐴𝐵)𝜓 ↔ ∃*𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844  wcel 2110  ∃*wmo 2540  ∃*wrmo 3069  cun 3890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-sb 2072  df-mo 2542  df-clab 2718  df-cleq 2732  df-clel 2818  df-rmo 3074  df-v 3433  df-un 3897
This theorem is referenced by:  disjxun0  30922
  Copyright terms: Public domain W3C validator