Theorem 2reurex 3766

Description: Double restricted quantification with existential uniqueness, analogous to 2euex 2641. (Contributed by Alexander van der Vekens, 24-Jun-2017.)

Assertion

Ref	Expression
2reurex	⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2reurex

Step	Hyp	Ref	Expression
1		reu5 3382	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑))
2		rexcom 3290	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
3		nfcv 2905	. . . . . 6 ⊢ Ⅎ𝑦𝐴
4		nfre1 3285	. . . . . 6 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐵 𝜑
5	3, 4	nfrmow 3413	. . . . 5 ⊢ Ⅎ𝑦∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑
6		rspe 3249	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑦 ∈ 𝐵 𝜑)
7	6	ex 412	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → ∃𝑦 ∈ 𝐵 𝜑))
8	7	ralrimivw 3150	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (𝜑 → ∃𝑦 ∈ 𝐵 𝜑))
9		rmoim 3746	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ∃𝑦 ∈ 𝐵 𝜑) → (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑))
10	8, 9	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑))
11	10	impcom 407	. . . . . . 7 ⊢ ((∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝑦 ∈ 𝐵) → ∃*𝑥 ∈ 𝐴 𝜑)
12		rmo5 3400	. . . . . . 7 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))
13	11, 12	sylib 218	. . . . . 6 ⊢ ((∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))
14	13	ex 412	. . . . 5 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)))
15	5, 14	reximdai 3261	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑))
16	2, 15	biimtrid 242	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑))
17	16	impcom 407	. 2 ⊢ ((∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑)
18	1, 17	sylbi 217	1 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  ∃!wreu 3378  ∃*wrmo 3379

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 1967  ax-7 2007  ax-8 2110  ax-10 2141  ax-11 2157  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-mo 2540  df-eu 2569  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381

This theorem is referenced by:  2rexreu  3768