Theorem 2euexv 2656

Description: Double quantification with existential uniqueness. Version of 2euex 2666 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2401. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.)

Assertion

Ref	Expression
2euexv	⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem 2euexv

Step	Hyp	Ref	Expression
1		df-eu 2594	. 2 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑))
2		excom 2199	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
3		nfe1 2187	. . . . . 6 ⊢ Ⅎ𝑦∃𝑦𝜑
4	3	nfmov 2585	. . . . 5 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑
5		19.8a 2217	. . . . . . 7 ⊢ (𝜑 → ∃𝑦𝜑)
6	5	moimi 2570	. . . . . 6 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑)
7		moeu 2608	. . . . . 6 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
8	6, 7	sylib 221	. . . . 5 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
9	4, 8	eximd 2252	. . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑦∃𝑥𝜑 → ∃𝑦∃!𝑥𝜑))
10	2, 9	biimtrid 245	. . 3 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑))
11	10	impcom 413	. 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → ∃𝑦∃!𝑥𝜑)
12	1, 11	sylbi 220	1 ⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1812  ∃*wmo 2562  ∃!weu 2593

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 2178  ax-11 2194  ax-12 2213

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 2564  df-eu 2594

This theorem is used by:  2exeuv  2657