Theorem 2exeu 2638

Description: Double existential uniqueness implies double unique existential quantification. The converse does not hold. Usage of this theorem is discouraged because it depends on ax-13 2367. Use the weaker 2exeuv 2624 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Assertion

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

Proof of Theorem 2exeu

Step	Hyp	Ref	Expression
1		eumo 2568	. . . 4 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃𝑦𝜑)
2		euex 2567	. . . . 5 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
3	2	moimi 2535	. . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
4	1, 3	syl 17	. . 3 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
5		2euex 2633	. . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∃𝑥∃!𝑦𝜑)
6	4, 5	anim12ci 613	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑))
7		df-eu 2559	. 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 1774  ∃*wmo 2528  ∃!weu 2558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2367

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-mo 2530  df-eu 2559

This theorem is referenced by:  2eu1  2642  2eu2  2644  2eu3  2645