Theorem eu2 1985

Description: An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)

Hypothesis

Ref	Expression
eu2.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
eu2	⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eu2

Step	Hyp	Ref	Expression
1		euex 1971	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
2		eu2.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
3	2	nfri 1452	. . . . 5 ⊢ (𝜑 → ∀𝑦𝜑)
4	3	eumo0 1972	. . . 4 ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
5	2	mo23 1982	. . . 4 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
6	4, 5	syl 14	. . 3 ⊢ (∃!𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
7	1, 6	jca 300	. 2 ⊢ (∃!𝑥𝜑 → (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
8		19.29r 1552	. . . 4 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∃𝑥(𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
9		impexp 259	. . . . . . . . 9 ⊢ (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ (𝜑 → ([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
10	9	albii 1399	. . . . . . . 8 ⊢ (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑦(𝜑 → ([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
11	2	19.21 1515	. . . . . . . 8 ⊢ (∀𝑦(𝜑 → ([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
12	10, 11	bitri 182	. . . . . . 7 ⊢ (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
13	12	anbi2i 444	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝜑 ∧ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦))))
14		abai 524	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) ↔ (𝜑 ∧ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦))))
15	13, 14	bitr4i 185	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
16	15	exbii 1536	. . . 4 ⊢ (∃𝑥(𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
17	8, 16	sylib 120	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
18	3	eu1 1966	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
19	17, 18	sylibr 132	. 2 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∃!𝑥𝜑)
20	7, 19	impbii 124	1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282  Ⅎwnf 1389  ∃wex 1421  [wsb 1685  ∃!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944

This theorem is referenced by:  eu3h  1986  mo3h  1994  bm1.1  2066  reu2  2780