Theorem 2eu7 2039

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 1427	. . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2	1	hbeu 1966	. . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∀𝑥∃!𝑦∃𝑥𝜑)
3	2	euan 2001	. 2 ⊢ (∃!𝑥(∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑))
4		ancom 262	. . . . 5 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃𝑦𝜑 ∧ ∃𝑥𝜑))
5	4	eubii 1954	. . . 4 ⊢ (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑))
6		hbe1 1427	. . . . 5 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
7	6	euan 2001	. . . 4 ⊢ (∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))
8		ancom 262	. . . 4 ⊢ ((∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑))
9	5, 7, 8	3bitri 204	. . 3 ⊢ (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑))
10	9	eubii 1954	. 2 ⊢ (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥(∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑))
11		ancom 262	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑))
12	3, 10, 11	3bitr4ri 211	1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))

Syntax hints:   ∧ wa 102   ↔ wb 103  ∃wex 1424  ∃!weu 1945

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949

This theorem is referenced by: (None)