Theorem 2exeuv 2627

Description: Double existential uniqueness implies double unique existential quantification. Version of 2exeu 2641 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2370. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2exeuv

Step	Hyp	Ref	Expression
1		eumo 2571	. . . 4 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃𝑦𝜑)
2		euex 2570	. . . . 5 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
3	2	moimi 2538	. . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
4	1, 3	syl 17	. . 3 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
5		2euexv 2626	. . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∃𝑥∃!𝑦𝜑)
6	4, 5	anim12ci 613	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑))
7		df-eu 2562	. 2 ⊢ (∃!𝑥∃!𝑦𝜑 ↔ (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑))
8	6, 7	sylibr 233	1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1780  ∃*wmo 2531  ∃!weu 2561

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 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-11 2153  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-mo 2533  df-eu 2562

This theorem is referenced by:  2eu1v  2646