Theorem rmounid 30266

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	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ¬ 𝜓)

Assertion

Ref	Expression
rmounid	⊢ (𝜑 → (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem rmounid

Step	Hyp	Ref	Expression
1		rmounid.1	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ¬ 𝜓)
2	1	ex 416	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → ¬ 𝜓))
3	2	con2d 136	. . . . . . . . . 10 ⊢ (𝜑 → (𝜓 → ¬ 𝑥 ∈ 𝐵))
4	3	imp 410	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑥 ∈ 𝐵)
5		biorf 934	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴)))
6		orcom 867	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴))
7	5, 6	syl6bbr 292	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)))
8	4, 7	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)))
9		elun 4076	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
10	8, 9	syl6bbr 292	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∪ 𝐵)))
11	10	pm5.32da 582	. . . . . 6 ⊢ (𝜑 → ((𝜓 ∧ 𝑥 ∈ 𝐴) ↔ (𝜓 ∧ 𝑥 ∈ (𝐴 ∪ 𝐵))))
12	11	biancomd 467	. . . . 5 ⊢ (𝜑 → ((𝜓 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜓)))
13	12	bicomd 226	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜓) ↔ (𝜓 ∧ 𝑥 ∈ 𝐴)))
14	13	biancomd 467	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)))
15	14	mobidv 2608	. 2 ⊢ (𝜑 → (∃*𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜓) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)))
16		df-rmo 3114	. 2 ⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜓 ↔ ∃*𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜓))
17		df-rmo 3114	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
18	15, 16, 17	3bitr4g 317	1 ⊢ (𝜑 → (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∈ wcel 2111  ∃*wmo 2596  ∃*wrmo 3109   ∪ cun 3879

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-mo 2598  df-clab 2777  df-cleq 2791  df-clel 2870  df-rmo 3114  df-v 3443  df-un 3886

This theorem is referenced by:  disjxun0  30337