Theorem bm1.1 1088

Description: Any set which has a property is the only set with that property. Theorem 1.1 of [BellMachover] p. 462.

Hypothesis

Ref	Expression
bm1.1.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
bm1.1	⊢ (∃x∀y(y ∈ x ↔ φ) → ∃!x∀y(y ∈ x ↔ φ))

Distinct variable group(s):   x,y

Proof of Theorem bm1.1

Step	Hyp	Ref	Expression
1		19.26 749	. . . . . 6 ⊢ (∀y((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) ↔ (∀y(y ∈ x ↔ φ) ∧ ∀y(y ∈ z ↔ φ)))
2		biantr 556	. . . . . . . 8 ⊢ (((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) → (y ∈ x ↔ y ∈ z))
3	2	19.20i 691	. . . . . . 7 ⊢ (∀y((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) → ∀y(y ∈ x ↔ y ∈ z))
4		ax-ext 1074	. . . . . . 7 ⊢ (∀y(y ∈ x ↔ y ∈ z) → x = z)
5	3, 4	syl 12	. . . . . 6 ⊢ (∀y((y ∈ x ↔ φ) ∧ (y ∈ z ↔ φ)) → x = z)
6	1, 5	sylbir 176	. . . . 5 ⊢ ((∀y(y ∈ x ↔ φ) ∧ ∀y(y ∈ z ↔ φ)) → x = z)
7		ax-17 925	. . . . . . . 8 ⊢ (y ∈ z → ∀x y ∈ z)
8		bm1.1.1	. . . . . . . 8 ⊢ (φ → ∀xφ)
9	7, 8	hbbi 705	. . . . . . 7 ⊢ ((y ∈ z ↔ φ) → ∀x(y ∈ z ↔ φ))
10	9	hbal 700	. . . . . 6 ⊢ (∀y(y ∈ z ↔ φ) → ∀x∀y(y ∈ z ↔ φ))
11		a14b 820	. . . . . . . 8 ⊢ (x = z → (y ∈ x ↔ y ∈ z))
12	11	bibi1d 471	. . . . . . 7 ⊢ (x = z → ((y ∈ x ↔ φ) ↔ (y ∈ z ↔ φ)))
13	12	bialdv 935	. . . . . 6 ⊢ (x = z → (∀y(y ∈ x ↔ φ) ↔ ∀y(y ∈ z ↔ φ)))
14	10, 13	sbie 904	. . . . 5 ⊢ ([z / x]∀y(y ∈ x ↔ φ) ↔ ∀y(y ∈ z ↔ φ))
15	6, 14	sylan2b 347	. . . 4 ⊢ ((∀y(y ∈ x ↔ φ) ∧ [z / x]∀y(y ∈ x ↔ φ)) → x = z)
16	15	gen2 681	. . 3 ⊢ ∀x∀z((∀y(y ∈ x ↔ φ) ∧ [z / x]∀y(y ∈ x ↔ φ)) → x = z)
17	16	jctr 239	. 2 ⊢ (∃x∀y(y ∈ x ↔ φ) → (∃x∀y(y ∈ x ↔ φ) ∧ ∀x∀z((∀y(y ∈ x ↔ φ) ∧ [z / x]∀y(y ∈ x ↔ φ)) → x = z)))
18		ax-17 925	. . 3 ⊢ (∀y(y ∈ x ↔ φ) → ∀z∀y(y ∈ x ↔ φ))
19	18	eu2 1023	. 2 ⊢ (∃!x∀y(y ∈ x ↔ φ) ↔ (∃x∀y(y ∈ x ↔ φ) ∧ ∀x∀z((∀y(y ∈ x ↔ φ) ∧ [z / x]∀y(y ∈ x ↔ φ)) → x = z)))
20	17, 19	sylibr 175	1 ⊢ (∃x∀y(y ∈ x ↔ φ) → ∃!x∀y(y ∈ x ↔ φ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803  [wsb 852  ∃!weu 1007

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-14 805  ax-16 922  ax-17 925  ax-ext 1074

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

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