Theorem rmounid 30257

Description: 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 415	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → ¬ 𝜓))
3	2	con2d 136	. . . . . . . . . 10 ⊢ (𝜑 → (𝜓 → ¬ 𝑥 ∈ 𝐵))
4	3	imp 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑥 ∈ 𝐵)
5		biorf 933	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴)))
6		orcom 866	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴))
7	5, 6	syl6bbr 291	. . . . . . . . 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 4118	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
10	8, 9	syl6bbr 291	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∪ 𝐵)))
11	10	pm5.32da 581	. . . . . 6 ⊢ (𝜑 → ((𝜓 ∧ 𝑥 ∈ 𝐴) ↔ (𝜓 ∧ 𝑥 ∈ (𝐴 ∪ 𝐵))))
12	11	biancomd 466	. . . . 5 ⊢ (𝜑 → ((𝜓 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜓)))
13	12	bicomd 225	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜓) ↔ (𝜓 ∧ 𝑥 ∈ 𝐴)))
14	13	biancomd 466	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)))
15	14	mobidv 2632	. 2 ⊢ (𝜑 → (∃*𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜓) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)))
16		df-rmo 3145	. 2 ⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜓 ↔ ∃*𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜓))
17		df-rmo 3145	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
18	15, 16, 17	3bitr4g 316	1 ⊢ (𝜑 → (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∈ wcel 2113  ∃*wmo 2619  ∃*wrmo 3140   ∪ cun 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rmo 3145  df-v 3493  df-un 3934

This theorem is referenced by:  disjxun0  30324