Theorem eu2 1023

Description: An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26.

Hypothesis

Ref	Expression
eu2.1	⊢ (φ → ∀yφ)

Assertion

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

Distinct variable group(s):   x,y

Proof of Theorem eu2

Step	Hyp	Ref	Expression
1		euex 1021	. . 3 ⊢ (∃!xφ → ∃xφ)
2		eu2.1	. . . . 5 ⊢ (φ → ∀yφ)
3	2	eumo0 1022	. . . 4 ⊢ (∃!xφ → ∃y∀x(φ → x = y))
4	2	mo 1020	. . . 4 ⊢ (∃y∀x(φ → x = y) ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))
5	3, 4	sylib 173	. . 3 ⊢ (∃!xφ → ∀x∀y((φ ∧ [y / x]φ) → x = y))
6	1, 5	jca 236	. 2 ⊢ (∃!xφ → (∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y)))
7		19.29r 753	. . . 4 ⊢ ((∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y)) → ∃x(φ ∧ ∀y((φ ∧ [y / x]φ) → x = y)))
8		impexp 276	. . . . . . . . 9 ⊢ (((φ ∧ [y / x]φ) → x = y) ↔ (φ → ([y / x]φ → x = y)))
9	8	bial 695	. . . . . . . 8 ⊢ (∀y((φ ∧ [y / x]φ) → x = y) ↔ ∀y(φ → ([y / x]φ → x = y)))
10	2	19.21 738	. . . . . . . 8 ⊢ (∀y(φ → ([y / x]φ → x = y)) ↔ (φ → ∀y([y / x]φ → x = y)))
11	9, 10	bitr 151	. . . . . . 7 ⊢ (∀y((φ ∧ [y / x]φ) → x = y) ↔ (φ → ∀y([y / x]φ → x = y)))
12	11	anbi2i 367	. . . . . 6 ⊢ ((φ ∧ ∀y((φ ∧ [y / x]φ) → x = y)) ↔ (φ ∧ (φ → ∀y([y / x]φ → x = y))))
13		abai 366	. . . . . 6 ⊢ ((φ ∧ ∀y([y / x]φ → x = y)) ↔ (φ ∧ (φ → ∀y([y / x]φ → x = y))))
14	12, 13	bitr4 154	. . . . 5 ⊢ ((φ ∧ ∀y((φ ∧ [y / x]φ) → x = y)) ↔ (φ ∧ ∀y([y / x]φ → x = y)))
15	14	biex 733	. . . 4 ⊢ (∃x(φ ∧ ∀y((φ ∧ [y / x]φ) → x = y)) ↔ ∃x(φ ∧ ∀y([y / x]φ → x = y)))
16	7, 15	sylib 173	. . 3 ⊢ ((∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y)) → ∃x(φ ∧ ∀y([y / x]φ → x = y)))
17	2	eu1 1019	. . 3 ⊢ (∃!xφ ↔ ∃x(φ ∧ ∀y([y / x]φ → x = y)))
18	16, 17	sylibr 175	. 2 ⊢ ((∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y)) → ∃!xφ)
19	6, 18	impbi 139	1 ⊢ (∃!xφ ↔ (∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  [wsb 852  ∃!weu 1007

This theorem is referenced by:  eu3 1024  bm1.1 1088  reu2 1338

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009

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