Theorem 2euex 1061

Description: Double quantification with existential uniqueness.

Assertion

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

Proof of Theorem 2euex

Step	Hyp	Ref	Expression
1		eu5 1035	. 2 ⊢ (∃!x∃yφ ↔ (∃x∃yφ ∧ ∃*x∃yφ))
2		hbe1 709	. . . . . . 7 ⊢ (∃yφ → ∀y∃yφ)
3	2	hbmo 1033	. . . . . 6 ⊢ (∃*x∃yφ → ∀y∃*x∃yφ)
4	3	19.41 774	. . . . 5 ⊢ (∃y(∃xφ ∧ ∃*x∃yφ) ↔ (∃y∃xφ ∧ ∃*x∃yφ))
5	4	biimpr 134	. . . 4 ⊢ ((∃y∃xφ ∧ ∃*x∃yφ) → ∃y(∃xφ ∧ ∃*x∃yφ))
6		excom 728	. . . 4 ⊢ (∃x∃yφ ↔ ∃y∃xφ)
7	5, 6	sylanb 344	. . 3 ⊢ ((∃x∃yφ ∧ ∃*x∃yφ) → ∃y(∃xφ ∧ ∃*x∃yφ))
8		2moex 1060	. . . . . . 7 ⊢ (∃*x∃yφ → ∀y∃*xφ)
9	8	19.21bi 742	. . . . . 6 ⊢ (∃*x∃yφ → ∃*xφ)
10	9	anim2i 270	. . . . 5 ⊢ ((∃xφ ∧ ∃*x∃yφ) → (∃xφ ∧ ∃*xφ))
11		eu5 1035	. . . . 5 ⊢ (∃!xφ ↔ (∃xφ ∧ ∃*xφ))
12	10, 11	sylibr 175	. . . 4 ⊢ ((∃xφ ∧ ∃*x∃yφ) → ∃!xφ)
13	12	19.22i 723	. . 3 ⊢ (∃y(∃xφ ∧ ∃*x∃yφ) → ∃y∃!xφ)
14	7, 13	syl 12	. 2 ⊢ ((∃x∃yφ ∧ ∃*x∃yφ) → ∃y∃!xφ)
15	1, 14	sylbi 174	1 ⊢ (∃!x∃yφ → ∃y∃!xφ)

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678  ∃!weu 1007  ∃*wmo 1008

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

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