Theorem 2eu1v 2736

Description: Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. Version of 2eu1 2735 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2390. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.)

Assertion

Ref	Expression
2eu1v	⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2eu1v

Step	Hyp	Ref	Expression
1		2eu2ex 2728	. . . . 5 ⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑)
2		moeu 2668	. . . . . . . 8 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
3	2	albii 1820	. . . . . . 7 ⊢ (∀𝑥∃*𝑦𝜑 ↔ ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑))
4		euim 2701	. . . . . . 7 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑)) → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑))
5	3, 4	sylan2b 595	. . . . . 6 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑))
6	5	ex 415	. . . . 5 ⊢ (∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑)))
7	1, 6	syl 17	. . . 4 ⊢ (∃!𝑥∃!𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑)))
8	7	pm2.43b 55	. . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑))
9		2euswapv 2715	. . . 4 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑))
10	8, 9	syld 47	. . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑦∃𝑥𝜑))
11	8, 10	jcad 515	. 2 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
12		2exeuv 2717	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
13	11, 12	impbid1 227	1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780  ∃*wmo 2620  ∃!weu 2653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654

This theorem is referenced by:  2eu5  2740