Theorem 2eu8 2560

Description: Two equivalent expressions for double existential uniqueness. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥∃!𝑦 using 2eu7 2559. (Contributed by NM, 20-Feb-2005.)

Assertion

Ref	Expression
2eu8	⊢ (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑))

Proof of Theorem 2eu8

Step	Hyp	Ref	Expression
1		2eu2 2554	. . 3 ⊢ (∃!𝑥∃𝑦𝜑 → (∃!𝑦∃!𝑥𝜑 ↔ ∃!𝑦∃𝑥𝜑))
2	1	pm5.32i 669	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))
3		nfeu1 2480	. . . . 5 ⊢ Ⅎ𝑥∃!𝑥𝜑
4	3	nfeu 2486	. . . 4 ⊢ Ⅎ𝑥∃!𝑦∃!𝑥𝜑
5	4	euan 2530	. . 3 ⊢ (∃!𝑥(∃!𝑦∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃!𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑))
6		ancom 466	. . . . . 6 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑥𝜑))
7	6	eubii 2492	. . . . 5 ⊢ (∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑦(∃𝑦𝜑 ∧ ∃!𝑥𝜑))
8		nfe1 2027	. . . . . 6 ⊢ Ⅎ𝑦∃𝑦𝜑
9	8	euan 2530	. . . . 5 ⊢ (∃!𝑦(∃𝑦𝜑 ∧ ∃!𝑥𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑))
10		ancom 466	. . . . 5 ⊢ ((∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑦∃!𝑥𝜑 ∧ ∃𝑦𝜑))
11	7, 9, 10	3bitri 286	. . . 4 ⊢ (∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃!𝑥𝜑 ∧ ∃𝑦𝜑))
12	11	eubii 2492	. . 3 ⊢ (∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥(∃!𝑦∃!𝑥𝜑 ∧ ∃𝑦𝜑))
13		ancom 466	. . 3 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑦∃!𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑))
14	5, 12, 13	3bitr4ri 293	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑))
15		2eu7 2559	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))
16	2, 14, 15	3bitr3ri 291	1 ⊢ (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑))

Syntax hints:   ↔ wb 196   ∧ wa 384  ∃wex 1704  ∃!weu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by: (None)