Theorem 2reurex 3726

Description: Double restricted quantification with existential uniqueness, analogous to 2euex 2671. (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 3372	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑))
2		rexcom 3294	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
3		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑦𝐴
4		nfre1 3290	. . . . . 6 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐵 𝜑
5	3, 4	nfrmow 3399	. . . . 5 ⊢ Ⅎ𝑦∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑
6		rspe 3255	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑦 ∈ 𝐵 𝜑)
7	6	ex 417	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → ∃𝑦 ∈ 𝐵 𝜑))
8	7	ralrimivw 3161	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (𝜑 → ∃𝑦 ∈ 𝐵 𝜑))
9		rmoim 3706	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ∃𝑦 ∈ 𝐵 𝜑) → (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑))
10	8, 9	syl 18	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑))
11	10	impcom 412	. . . . . . 7 ⊢ ((∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝑦 ∈ 𝐵) → ∃*𝑥 ∈ 𝐴 𝜑)
12		rmo5 3388	. . . . . . 7 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))
13	11, 12	sylib 221	. . . . . 6 ⊢ ((∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))
14	13	ex 417	. . . . 5 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)))
15	5, 14	reximdai 3267	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑))
16	2, 15	biimtrid 245	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑))
17	16	impcom 412	. 2 ⊢ ((∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑)
18	1, 17	sylbi 220	1 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  ∃!wreu 3368  ∃*wrmo 3369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-10 2178  ax-11 2194  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-mo 2569  df-eu 2599  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371

This theorem is referenced by:  2rexreu  3728