Theorem 2eu7 1448

Description: Two equivalent expressions for double existential uniqueness.

Assertion

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

Proof of Theorem 2eu7

Step	Hyp	Ref	Expression
1		hbe1 1012	. . . 4 ⊢ (∃xφ → ∀x∃xφ)
2	1	hbeu 1382	. . 3 ⊢ (∃!y∃xφ → ∀x∃!y∃xφ)
3	2	euan 1421	. 2 ⊢ (∃!x(∃!y∃xφ ⋀ ∃yφ) ↔ (∃!y∃xφ ⋀ ∃!x∃yφ))
4		ancom 435	. . . . 5 ⊢ ((∃xφ ⋀ ∃yφ) ↔ (∃yφ ⋀ ∃xφ))
5	4	eubii 1380	. . . 4 ⊢ (∃!y(∃xφ ⋀ ∃yφ) ↔ ∃!y(∃yφ ⋀ ∃xφ))
6		hbe1 1012	. . . . 5 ⊢ (∃yφ → ∀y∃yφ)
7	6	euan 1421	. . . 4 ⊢ (∃!y(∃yφ ⋀ ∃xφ) ↔ (∃yφ ⋀ ∃!y∃xφ))
8		ancom 435	. . . 4 ⊢ ((∃yφ ⋀ ∃!y∃xφ) ↔ (∃!y∃xφ ⋀ ∃yφ))
9	5, 7, 8	3bitr 177	. . 3 ⊢ (∃!y(∃xφ ⋀ ∃yφ) ↔ (∃!y∃xφ ⋀ ∃yφ))
10	9	eubii 1380	. 2 ⊢ (∃!x∃!y(∃xφ ⋀ ∃yφ) ↔ ∃!x(∃!y∃xφ ⋀ ∃yφ))
11		ancom 435	. 2 ⊢ ((∃!x∃yφ ⋀ ∃!y∃xφ) ↔ (∃!y∃xφ ⋀ ∃!x∃yφ))
12	3, 10, 11	3bitr4r 184	1 ⊢ ((∃!x∃yφ ⋀ ∃!y∃xφ) ↔ ∃!x∃!y(∃xφ ⋀ ∃yφ))

Syntax hints:   ↔ wb 146   ⋀ wa 223  ∃wex 977  ∃!weu 1373

This theorem is referenced by:  2eu8 1449

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376

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