Theorem 2eu7 2687

Description: Two equivalent expressions for double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2406. (Contributed by NM, 19-Feb-2005.) (New usage is discouraged.)

Assertion

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

Proof of Theorem 2eu7

Step	Hyp	Ref	Expression
1		nfe1 2188	. . . 4 ⊢ Ⅎ𝑥∃𝑥𝜑
2	1	nfeu 2624	. . 3 ⊢ Ⅎ𝑥∃!𝑦∃𝑥𝜑
3	2	euan 2651	. 2 ⊢ (∃!𝑥(∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑))
4		ancom 466	. . . . 5 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃𝑦𝜑 ∧ ∃𝑥𝜑))
5	4	eubii 2615	. . . 4 ⊢ (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑))
6		nfe1 2188	. . . . 5 ⊢ Ⅎ𝑦∃𝑦𝜑
7	6	euan 2651	. . . 4 ⊢ (∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))
8		ancom 466	. . . 4 ⊢ ((∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑))
9	5, 7, 8	3bitri 300	. . 3 ⊢ (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑))
10	9	eubii 2615	. 2 ⊢ (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥(∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑))
11		ancom 466	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑))
12	3, 10, 11	3bitr4ri 307	1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1812  ∃!weu 2598

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-mo 2569  df-eu 2599

This theorem is used by:  2eu8  2688