Theorem 2eu7 2181

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 1548	. . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2	1	hbeu 2107	. . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∀𝑥∃!𝑦∃𝑥𝜑)
3	2	euan 2143	. 2 ⊢ (∃!𝑥(∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑))
4		ancom 266	. . . . 5 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃𝑦𝜑 ∧ ∃𝑥𝜑))
5	4	eubii 2095	. . . 4 ⊢ (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑))
6		hbe1 1548	. . . . 5 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
7	6	euan 2143	. . . 4 ⊢ (∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))
8		ancom 266	. . . 4 ⊢ ((∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑))
9	5, 7, 8	3bitri 206	. . 3 ⊢ (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑))
10	9	eubii 2095	. 2 ⊢ (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥(∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑))
11		ancom 266	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑))
12	3, 10, 11	3bitr4ri 213	1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1545  ∃!weu 2086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090

This theorem is referenced by: (None)