Theorem 2reu8 44604

Description: Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2660. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥 ∈ 𝐴∃!𝑦 ∈ 𝐵 using 2reu7 44603. (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 3831	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
2	1	pm5.32i 575	. 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
3		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑥𝐵
4		nfreu1 3300	. . . . 5 ⊢ Ⅎ𝑥∃!𝑥 ∈ 𝐴 𝜑
5	3, 4	nfreuw 3305	. . . 4 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑
6	5	reuan 3829	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 (∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑))
7		ancom 461	. . . . . 6 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜑))
8	7	reubii 3325	. . . . 5 ⊢ (∃!𝑦 ∈ 𝐵 (∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ ∃!𝑦 ∈ 𝐵 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜑))
9		nfre1 3239	. . . . . 6 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐵 𝜑
10	9	reuan 3829	. . . . 5 ⊢ (∃!𝑦 ∈ 𝐵 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑))
11		ancom 461	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))
12	8, 10, 11	3bitri 297	. . . 4 ⊢ (∃!𝑦 ∈ 𝐵 (∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))
13	12	reubii 3325	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ ∃!𝑥 ∈ 𝐴 (∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))
14		ancom 461	. . 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 304	. 2 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑) ↔ ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))
16		2reu7 44603	. 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 302	1 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 (∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜑))

Syntax hints:   ↔ wb 205   ∧ wa 396  ∃wrex 3065  ∃!wreu 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-mo 2540  df-eu 2569  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072

This theorem is referenced by: (None)