Theorem eu1 1925

Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.)

Hypothesis

Ref	Expression
eu1.1	⊢ (𝜑 → ∀𝑦𝜑)

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		hbs1 1814	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
2	1	euf 1905	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑥∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑥))
3		eu1.1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
4	3	sb8euh 1923	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
5		equcom 1593	. . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
6	5	imbi2i 215	. . . . . 6 ⊢ (([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑥))
7	6	albii 1359	. . . . 5 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑥))
8	3	sb6rf 1733	. . . . 5 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
9	7, 8	anbi12i 433	. . . 4 ⊢ ((∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦) ∧ 𝜑) ↔ (∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑥) ∧ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)))
10		ancom 253	. . . 4 ⊢ ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) ↔ (∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦) ∧ 𝜑))
11		albiim 1376	. . . 4 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑥) ↔ (∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑥) ∧ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)))
12	9, 10, 11	3bitr4i 201	. . 3 ⊢ ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑥))
13	12	exbii 1496	. 2 ⊢ (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) ↔ ∃𝑥∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑥))
14	2, 4, 13	3bitr4i 201	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  ∃wex 1381  [wsb 1645  ∃!weu 1900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903

This theorem is referenced by:  euex  1930  eu2  1944