Theorem axextmo 2808

Description: There exists at most one set with prescribed elements. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Wolf Lammen, 13-Nov-2019.) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022.)

Hypothesis

Ref	Expression
axextmo.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
axextmo	⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem axextmo

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		biantr 840	. . . . 5 ⊢ (((𝑦 ∈ 𝑥 ↔ 𝜑) ∧ (𝑦 ∈ 𝑧 ↔ 𝜑)) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
2	1	alanimi 1915	. . . 4 ⊢ ((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
3		ax-ext 2803	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧) → 𝑥 = 𝑧)
4	2, 3	syl 17	. . 3 ⊢ ((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → 𝑥 = 𝑧)
5	4	gen2 1895	. 2 ⊢ ∀𝑥∀𝑧((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → 𝑥 = 𝑧)
6		nfv 2013	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
7		axextmo.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
8	6, 7	nfbi 2006	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑧 ↔ 𝜑)
9	8	nfal 2355	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)
10		elequ2 2178	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
11	10	bibi1d 335	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑦 ∈ 𝑥 ↔ 𝜑) ↔ (𝑦 ∈ 𝑧 ↔ 𝜑)))
12	11	albidv 2019	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)))
13	9, 12	mo4f 2636	. 2 ⊢ (∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ↔ ∀𝑥∀𝑧((∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) ∧ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝜑)) → 𝑥 = 𝑧))
14	5, 13	mpbir 223	1 ⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1654  Ⅎwnf 1882  ∃*wmo 2603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605

This theorem is referenced by:  nulmo  2810  zfnuleuOLD  5012