Theorem eu2 2229

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 2227	. . 3 ⊢ (∃!xφ → ∃xφ)
2		eu2.1	. . . . 5 ⊢ Ⅎyφ
3	2	eumo0 2228	. . . 4 ⊢ (∃!xφ → ∃y∀x(φ → x = y))
4	2	mo 2226	. . . 4 ⊢ (∃y∀x(φ → x = y) ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))
5	3, 4	sylib 188	. . 3 ⊢ (∃!xφ → ∀x∀y((φ ∧ [y / x]φ) → x = y))
6	1, 5	jca 518	. 2 ⊢ (∃!xφ → (∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y)))
7		19.29r 1597	. . . 4 ⊢ ((∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y)) → ∃x(φ ∧ ∀y((φ ∧ [y / x]φ) → x = y)))
8		impexp 433	. . . . . . . . 9 ⊢ (((φ ∧ [y / x]φ) → x = y) ↔ (φ → ([y / x]φ → x = y)))
9	8	albii 1566	. . . . . . . 8 ⊢ (∀y((φ ∧ [y / x]φ) → x = y) ↔ ∀y(φ → ([y / x]φ → x = y)))
10	2	19.21 1796	. . . . . . . 8 ⊢ (∀y(φ → ([y / x]φ → x = y)) ↔ (φ → ∀y([y / x]φ → x = y)))
11	9, 10	bitri 240	. . . . . . 7 ⊢ (∀y((φ ∧ [y / x]φ) → x = y) ↔ (φ → ∀y([y / x]φ → x = y)))
12	11	anbi2i 675	. . . . . 6 ⊢ ((φ ∧ ∀y((φ ∧ [y / x]φ) → x = y)) ↔ (φ ∧ (φ → ∀y([y / x]φ → x = y))))
13		abai 770	. . . . . 6 ⊢ ((φ ∧ ∀y([y / x]φ → x = y)) ↔ (φ ∧ (φ → ∀y([y / x]φ → x = y))))
14	12, 13	bitr4i 243	. . . . 5 ⊢ ((φ ∧ ∀y((φ ∧ [y / x]φ) → x = y)) ↔ (φ ∧ ∀y([y / x]φ → x = y)))
15	14	exbii 1582	. . . 4 ⊢ (∃x(φ ∧ ∀y((φ ∧ [y / x]φ) → x = y)) ↔ ∃x(φ ∧ ∀y([y / x]φ → x = y)))
16	7, 15	sylib 188	. . 3 ⊢ ((∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y)) → ∃x(φ ∧ ∀y([y / x]φ → x = y)))
17	2	eu1 2225	. . 3 ⊢ (∃!xφ ↔ ∃x(φ ∧ ∀y([y / x]φ → x = y)))
18	16, 17	sylibr 203	. 2 ⊢ ((∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y)) → ∃!xφ)
19	6, 18	impbii 180	1 ⊢ (∃!xφ ↔ (∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  [wsb 1648  ∃!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208

This theorem is referenced by:  eu3  2230  bm1.1  2338  reu2  3025