Theorem bm1.1 1460

Description: Any set defined by a property is the only set defined by 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:   x,y

Proof of Theorem bm1.1

Step	Hyp	Ref	Expression
1		19.26 1065	. . . . . 6 ⊢ (∀y((y ∈ x ↔ φ) ⋀ (y ∈ z ↔ φ)) ↔ (∀y(y ∈ x ↔ φ) ⋀ ∀y(y ∈ z ↔ φ)))
2		biantr 741	. . . . . . . 8 ⊢ (((y ∈ x ↔ φ) ⋀ (y ∈ z ↔ φ)) → (y ∈ x ↔ y ∈ z))
3	2	19.20i 990	. . . . . . 7 ⊢ (∀y((y ∈ x ↔ φ) ⋀ (y ∈ z ↔ φ)) → ∀y(y ∈ x ↔ y ∈ z))
4		ax-ext 1457	. . . . . . 7 ⊢ (∀y(y ∈ x ↔ y ∈ z) → x = z)
5	3, 4	syl 10	. . . . . 6 ⊢ (∀y((y ∈ x ↔ φ) ⋀ (y ∈ z ↔ φ)) → x = z)
6	1, 5	sylbir 201	. . . . 5 ⊢ ((∀y(y ∈ x ↔ φ) ⋀ ∀y(y ∈ z ↔ φ)) → x = z)
7		ax-17 969	. . . . . . . 8 ⊢ (y ∈ z → ∀x y ∈ z)
8		bm1.1.1	. . . . . . . 8 ⊢ (φ → ∀xφ)
9	7, 8	hbbi 1008	. . . . . . 7 ⊢ ((y ∈ z ↔ φ) → ∀x(y ∈ z ↔ φ))
10	9	hbal 1003	. . . . . 6 ⊢ (∀y(y ∈ z ↔ φ) → ∀x∀y(y ∈ z ↔ φ))
11		elequ2 1135	. . . . . . . 8 ⊢ (x = z → (y ∈ x ↔ y ∈ z))
12	11	bibi1d 618	. . . . . . 7 ⊢ (x = z → ((y ∈ x ↔ φ) ↔ (y ∈ z ↔ φ)))
13	12	albidv 1276	. . . . . 6 ⊢ (x = z → (∀y(y ∈ x ↔ φ) ↔ ∀y(y ∈ z ↔ φ)))
14	10, 13	sbie 1194	. . . . 5 ⊢ ([z / x]∀y(y ∈ x ↔ φ) ↔ ∀y(y ∈ z ↔ φ))
15	6, 14	sylan2b 452	. . . 4 ⊢ ((∀y(y ∈ x ↔ φ) ⋀ [z / x]∀y(y ∈ x ↔ φ)) → x = z)
16	15	gen2 981	. . 3 ⊢ ∀x∀z((∀y(y ∈ x ↔ φ) ⋀ [z / x]∀y(y ∈ x ↔ φ)) → x = z)
17	16	jctr 291	. 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-17 969	. . 3 ⊢ (∀y(y ∈ x ↔ φ) → ∀z∀y(y ∈ x ↔ φ))
19	18	eu2 1394	. 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 200	1 ⊢ (∃x∀y(y ∈ x ↔ φ) → ∃!x∀y(y ∈ x ↔ φ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 952   = wceq 954   ∈ wcel 956  ∃wex 978  [wsbc 1168  ∃!weu 1378

This theorem is referenced by:  zfnuleu 2702

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380

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