Theorem 2euex 2671

Description: Double quantification with existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2406. Use the weaker 2euexv 2661 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.)

Assertion

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

Proof of Theorem 2euex

Step	Hyp	Ref	Expression
1		df-eu 2599	. 2 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑))
2		excom 2200	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
3		nfe1 2188	. . . . . 6 ⊢ Ⅎ𝑦∃𝑦𝜑
4	3	nfmo 2592	. . . . 5 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑
5		19.8a 2220	. . . . . . 7 ⊢ (𝜑 → ∃𝑦𝜑)
6	5	moimi 2575	. . . . . 6 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑)
7		moeu 2613	. . . . . 6 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
8	6, 7	sylib 221	. . . . 5 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
9	4, 8	eximd 2255	. . . 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 2567  ∃!weu 2598

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 2179  ax-11 2195  ax-12 2216  ax-13 2406

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 2569  df-eu 2599

This theorem is used by:  2exeu  2676