Theorem 2eu3 2679

Description: Double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2402. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 23-Apr-2023.) (New usage is discouraged.)

Assertion

Ref	Expression
2eu3	⊢ (∀𝑥∀𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))

Proof of Theorem 2eu3

Step	Hyp	Ref	Expression
1		nfmo1 2583	. . . . 5 ⊢ Ⅎ𝑦∃*𝑦𝜑
2	1	19.31 2268	. . . 4 ⊢ (∀𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑))
3	2	albii 1838	. . 3 ⊢ (∀𝑥∀𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ ∀𝑥(∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑))
4		nfmo1 2583	. . . . 5 ⊢ Ⅎ𝑥∃*𝑥𝜑
5	4	nfal 2354	. . . 4 ⊢ Ⅎ𝑥∀𝑦∃*𝑥𝜑
6	5	19.32 2267	. . 3 ⊢ (∀𝑥(∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑))
7	3, 6	bitri 277	. 2 ⊢ (∀𝑥∀𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑))
8		2eu1 2676	. . . . . . 7 ⊢ (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑)))
9	8	biimpd 231	. . . . . 6 ⊢ (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 → (∃!𝑦∃𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑)))
10		ancom 464	. . . . . 6 ⊢ ((∃!𝑦∃𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑) ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))
11	9, 10	imbitrdi 253	. . . . 5 ⊢ (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 → (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
12		2eu1 2676	. . . . . 6 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
13	12	biimpd 231	. . . . 5 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
14	11, 13	jaoa 968	. . . 4 ⊢ ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑦∃!𝑥𝜑 ∧ ∃!𝑥∃!𝑦𝜑) → (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
15	14	ancomsd 469	. . 3 ⊢ ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) → (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
16		2exeu 2672	. . . 4 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
17		2exeu 2672	. . . . 5 ⊢ ((∃!𝑦∃𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑) → ∃!𝑦∃!𝑥𝜑)
18	17	ancoms 462	. . . 4 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑦∃!𝑥𝜑)
19	16, 18	jca 519	. . 3 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → (∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑))
20	15, 19	impbid1 227	. 2 ⊢ ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
21	7, 20	sylbi 219	1 ⊢ (∀𝑥∀𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858  ∀wal 1557  ∃wex 1798  ∃*wmo 2563  ∃!weu 2594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-10 2174  ax-11 2190  ax-12 2211  ax-13 2402

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-mo 2565  df-eu 2595

This theorem is referenced by: (None)