Theorem 2euex 2672

Description: Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

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

Proof of Theorem 2euex

Step	Hyp	Ref	Expression
1		df-eu 2584	. 2 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑))
2		excom 2098	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
3		nfe1 2087	. . . . . 6 ⊢ Ⅎ𝑦∃𝑦𝜑
4	3	nfmo 2574	. . . . 5 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑
5		19.8a 2109	. . . . . . 7 ⊢ (𝜑 → ∃𝑦𝜑)
6	5	moimi 2553	. . . . . 6 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑)
7		moeu 2602	. . . . . 6 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
8	6, 7	sylib 210	. . . . 5 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
9	4, 8	eximd 2146	. . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑦∃𝑥𝜑 → ∃𝑦∃!𝑥𝜑))
10	2, 9	syl5bi 234	. . 3 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑))
11	10	impcom 399	. 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → ∃𝑦∃!𝑥𝜑)
12	1, 11	sylbi 209	1 ⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 387  ∃wex 1742  ∃*wmo 2545  ∃!weu 2583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-mo 2547  df-eu 2584

This theorem is referenced by:  2exeu  2677