Theorem 2eu1 2707

Description: Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 23-Apr-2023.)

Assertion

Ref	Expression
2eu1	⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))

Proof of Theorem 2eu1

Step	Hyp	Ref	Expression
1		2eu2ex 2700	. . . . 5 ⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑)
2		moeu 2630	. . . . . . . 8 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
3	2	albii 1805	. . . . . . 7 ⊢ (∀𝑥∃*𝑦𝜑 ↔ ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑))
4		euim 2671	. . . . . . 7 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑)) → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑))
5	3, 4	sylan2b 593	. . . . . 6 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑))
6	5	ex 413	. . . . 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		2euswap 2702	. . . 4 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑))
10	8, 9	syld 47	. . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑦∃𝑥𝜑))
11	8, 10	jcad 513	. 2 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
12		2exeu 2703	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
13	11, 12	impbid1 226	1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1523  ∃wex 1765  ∃*wmo 2576  ∃!weu 2613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614

This theorem is referenced by:  2eu2  2710  2eu3  2711  2eu3OLD  2712  2eu5OLD  2715