Theorem 2euex 2028

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		eu5 1988	. 2 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑))
2		excom 1594	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
3		hbe1 1424	. . . . . 6 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
4	3	hbmo 1980	. . . . 5 ⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥∃𝑦𝜑)
5		19.8a 1522	. . . . . . 7 ⊢ (𝜑 → ∃𝑦𝜑)
6	5	moimi 2006	. . . . . 6 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑)
7		df-mo 1945	. . . . . 6 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
8	6, 7	sylib 120	. . . . 5 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
9	4, 8	eximdh 1542	. . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑦∃𝑥𝜑 → ∃𝑦∃!𝑥𝜑))
10	2, 9	syl5bi 150	. . 3 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑))
11	10	impcom 123	. 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → ∃𝑦∃!𝑥𝜑)
12	1, 11	sylbi 119	1 ⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 102  ∃wex 1421  ∃!weu 1941  ∃*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by: (None)