Theorem 2eu7 2113

Description: Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)

Assertion

Ref	Expression
2eu7	⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))

Proof of Theorem 2eu7

Step	Hyp	Ref	Expression
1		hbe1 1488	. . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2	1	hbeu 2040	. . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∀𝑥∃!𝑦∃𝑥𝜑)
3	2	euan 2075	. 2 ⊢ (∃!𝑥(∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑))
4		ancom 264	. . . . 5 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃𝑦𝜑 ∧ ∃𝑥𝜑))
5	4	eubii 2028	. . . 4 ⊢ (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑))
6		hbe1 1488	. . . . 5 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
7	6	euan 2075	. . . 4 ⊢ (∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))
8		ancom 264	. . . 4 ⊢ ((∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑))
9	5, 7, 8	3bitri 205	. . 3 ⊢ (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑))
10	9	eubii 2028	. 2 ⊢ (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥(∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑))
11		ancom 264	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑))
12	3, 10, 11	3bitr4ri 212	1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1485  ∃!weu 2019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023

This theorem is referenced by: (None)