Theorem eu1 2608

Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Oct-2018.) Avoid ax-13 2375. (Revised by Wolf Lammen, 7-Feb-2023.)

Hypothesis

Ref	Expression
eu1.nf	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
eu1	⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eu1

Step	Hyp	Ref	Expression
1		nfs1v 2155	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
2	1	euf 2574	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑥∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑥))
3		eu1.nf	. . 3 ⊢ Ⅎ𝑦𝜑
4	3	sb8euv 2597	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
5	3	sb6rfv 2358	. . . . 5 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
6		equcom 2016	. . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
7	6	imbi2i 336	. . . . . 6 ⊢ (([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑥))
8	7	albii 1818	. . . . 5 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑥))
9	5, 8	anbi12ci 629	. . . 4 ⊢ ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) ↔ (∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑥) ∧ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)))
10		albiim 1888	. . . 4 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑥) ↔ (∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑥) ∧ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)))
11	9, 10	bitr4i 278	. . 3 ⊢ ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑥))
12	11	exbii 1847	. 2 ⊢ (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) ↔ ∃𝑥∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑥))
13	2, 4, 12	3bitr4i 303	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1537  ∃wex 1778  Ⅎwnf 1782  [wsb 2063  ∃!weu 2566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-11 2156  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567

This theorem is referenced by:  kmlem15  10171