Theorem 2reu8 47890

Description: Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2689. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥 ∈ 𝐴∃!𝑦 ∈ 𝐵 using 2reu7 47889. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Assertion

Ref	Expression
2reu8	⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))

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

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem 2reu8

Step	Hyp	Ref	Expression
1		2reu2 3855	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
2	1	pm5.32i 585	. 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
3		nfcv 2928	. . . . 5 ⊢ Ⅎ𝑥𝐵
4		nfreu1 3400	. . . . 5 ⊢ Ⅎ𝑥∃!𝑥 ∈ 𝐴 𝜑
5	3, 4	nfreuw 3402	. . . 4 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑
6	5	reuan 3853	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 (∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑))
7		ancom 466	. . . . . 6 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜑))
8	7	reubii 3381	. . . . 5 ⊢ (∃!𝑦 ∈ 𝐵 (∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ ∃!𝑦 ∈ 𝐵 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜑))
9		nfre1 3293	. . . . . 6 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐵 𝜑
10	9	reuan 3853	. . . . 5 ⊢ (∃!𝑦 ∈ 𝐵 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑))
11		ancom 466	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))
12	8, 10, 11	3bitri 300	. . . 4 ⊢ (∃!𝑦 ∈ 𝐵 (∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))
13	12	reubii 3381	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ ∃!𝑥 ∈ 𝐴 (∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))
14		ancom 466	. . 3 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑))
15	6, 13, 14	3bitr4ri 307	. 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑) ↔ ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))
16		2reu7 47889	. 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))
17	2, 15, 16	3bitr3ri 305	1 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wrex 3092  ∃!wreu 3370

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-mo 2570  df-eu 2600  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373

This theorem is used by: (None)