Theorem hbeu 1016

Description: Bound-variable hypothesis builder for "at most one". Note that x and y needn't be distinct (this makes the proof more difficult).

Hypothesis

Ref	Expression
hbeu.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
hbeu	⊢ (∃!yφ → ∀x∃!yφ)

Proof of Theorem hbeu

Step	Hyp	Ref	Expression
1		ax-10 800	. . . . . 6 ⊢ (∀y y = x → (∀y∀y(φ ↔ y = z) → ∀x∀y(φ ↔ y = z)))
2	1	eq4s 822	. . . . 5 ⊢ (∀x x = y → (∀y∀y(φ ↔ y = z) → ∀x∀y(φ ↔ y = z)))
3		hba1 698	. . . . 5 ⊢ (∀y(φ ↔ y = z) → ∀y∀y(φ ↔ y = z))
4	2, 3	syl5 22	. . . 4 ⊢ (∀x x = y → (∀y(φ ↔ y = z) → ∀x∀y(φ ↔ y = z)))
5		eq6 826	. . . . 5 ⊢ (¬ ∀x x = y → ∀y ¬ ∀x x = y)
6		eq6 826	. . . . . 6 ⊢ (¬ ∀x x = y → ∀x ¬ ∀x x = y)
7		hbeu.1	. . . . . . 7 ⊢ (φ → ∀xφ)
8	7	a1i 7	. . . . . 6 ⊢ (¬ ∀x x = y → (φ → ∀xφ))
9		ddeeq1 1001	. . . . . 6 ⊢ (¬ ∀x x = y → (y = z → ∀x y = z))
10	6, 8, 9	hbbid 789	. . . . 5 ⊢ (¬ ∀x x = y → ((φ ↔ y = z) → ∀x(φ ↔ y = z)))
11	5, 10	hbald 790	. . . 4 ⊢ (¬ ∀x x = y → (∀y(φ ↔ y = z) → ∀x∀y(φ ↔ y = z)))
12	4, 11	pm2.61i 110	. . 3 ⊢ (∀y(φ ↔ y = z) → ∀x∀y(φ ↔ y = z))
13	12	hbex 701	. 2 ⊢ (∃z∀y(φ ↔ y = z) → ∀x∃z∀y(φ ↔ y = z))
14		df-eu 1009	. 2 ⊢ (∃!yφ ↔ ∃z∀y(φ ↔ y = z))
15	14	bial 695	. 2 ⊢ (∀x∃!yφ ↔ ∀x∃z∀y(φ ↔ y = z))
16	13, 14, 15	3imtr4 192	1 ⊢ (∃!yφ → ∀x∃!yφ)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678   = weq 797  ∃!weu 1007

This theorem is referenced by:  hbmo 1033

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-17 925

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

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