Theorem eu2 2122

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 2107	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
2		eu2.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
3	2	nfri 1565	. . . . 5 ⊢ (𝜑 → ∀𝑦𝜑)
4	3	eumo0 2108	. . . 4 ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
5	2	mo23 2119	. . . 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 306	. 2 ⊢ (∃!𝑥𝜑 → (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
8		19.29r 1667	. . . 4 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∃𝑥(𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
9		impexp 263	. . . . . . . . 9 ⊢ (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ (𝜑 → ([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
10	9	albii 1516	. . . . . . . 8 ⊢ (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑦(𝜑 → ([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
11	2	19.21 1629	. . . . . . . 8 ⊢ (∀𝑦(𝜑 → ([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
12	10, 11	bitri 184	. . . . . . 7 ⊢ (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
13	12	anbi2i 457	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝜑 ∧ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦))))
14		abai 560	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) ↔ (𝜑 ∧ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦))))
15	13, 14	bitr4i 187	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
16	15	exbii 1651	. . . 4 ⊢ (∃𝑥(𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
17	8, 16	sylib 122	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
18	3	eu1 2102	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
19	17, 18	sylibr 134	. 2 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∃!𝑥𝜑)
20	7, 19	impbii 126	1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1393  Ⅎwnf 1506  ∃wex 1538  [wsb 1808  ∃!weu 2077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080

This theorem is referenced by:  eu3h  2123  mo3h  2131  bm1.1  2214  reu2  2991