Theorem 2exeuv 2630

Description: Double existential uniqueness implies double unique existential quantification. Version of 2exeu 2644 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2375. (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 2576	. . . 4 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃𝑦𝜑)
2		euex 2575	. . . . 5 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
3	2	moimi 2543	. . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
4	1, 3	syl 17	. . 3 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
5		2euexv 2629	. . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∃𝑥∃!𝑦𝜑)
6	4, 5	anim12ci 614	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑))
7		df-eu 2567	. 2 ⊢ (∃!𝑥∃!𝑦𝜑 ↔ (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑))
8	6, 7	sylibr 234	1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1776  ∃*wmo 2536  ∃!weu 2566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-mo 2538  df-eu 2567

This theorem is referenced by:  2eu1v  2650