Theorem 2euex 2140

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 2100	. 2 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑))
2		excom 1686	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
3		hbe1 1517	. . . . . 6 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
4	3	hbmo 2092	. . . . 5 ⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥∃𝑦𝜑)
5		19.8a 1612	. . . . . . 7 ⊢ (𝜑 → ∃𝑦𝜑)
6	5	moimi 2118	. . . . . 6 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑)
7		df-mo 2057	. . . . . 6 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
8	6, 7	sylib 122	. . . . 5 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
9	4, 8	eximdh 1633	. . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑦∃𝑥𝜑 → ∃𝑦∃!𝑥𝜑))
10	2, 9	biimtrid 152	. . 3 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑))
11	10	impcom 125	. 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → ∃𝑦∃!𝑥𝜑)
12	1, 11	sylbi 121	1 ⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1514  ∃!weu 2053  ∃*wmo 2054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057

This theorem is referenced by: (None)