Theorem eu1 1019

Description: An alternate way of expressing uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110.

Hypothesis

Ref	Expression
eu1.1	⊢ (φ → ∀yφ)

Assertion

Ref	Expression
eu1	⊢ (∃!xφ ↔ ∃x(φ ∧ ∀y([y / x]φ → x = y)))

Distinct variable group(s):   x,y

Proof of Theorem eu1

Step	Hyp	Ref	Expression
1		hbs1 986	. . 3 ⊢ ([y / x]φ → ∀x[y / x]φ)
2	1	euf 1011	. 2 ⊢ (∃!y[y / x]φ ↔ ∃x∀y([y / x]φ ↔ y = x))
3		eu1.1	. . 3 ⊢ (φ → ∀yφ)
4	3	sb8eu 1017	. 2 ⊢ (∃!xφ ↔ ∃!y[y / x]φ)
5		eqcomb 812	. . . . . . 7 ⊢ (x = y ↔ y = x)
6	5	imbi2i 160	. . . . . 6 ⊢ (([y / x]φ → x = y) ↔ ([y / x]φ → y = x))
7	6	bial 695	. . . . 5 ⊢ (∀y([y / x]φ → x = y) ↔ ∀y([y / x]φ → y = x))
8	3	sb5f1 917	. . . . 5 ⊢ (φ ↔ ∀y(y = x → [y / x]φ))
9	7, 8	anbi12i 369	. . . 4 ⊢ ((∀y([y / x]φ → x = y) ∧ φ) ↔ (∀y([y / x]φ → y = x) ∧ ∀y(y = x → [y / x]φ)))
10		ancom 333	. . . 4 ⊢ ((φ ∧ ∀y([y / x]φ → x = y)) ↔ (∀y([y / x]φ → x = y) ∧ φ))
11		albi 785	. . . 4 ⊢ (∀y([y / x]φ ↔ y = x) ↔ (∀y([y / x]φ → y = x) ∧ ∀y(y = x → [y / x]φ)))
12	9, 10, 11	3bitr4 158	. . 3 ⊢ ((φ ∧ ∀y([y / x]φ → x = y)) ↔ ∀y([y / x]φ ↔ y = x))
13	12	biex 733	. 2 ⊢ (∃x(φ ∧ ∀y([y / x]φ → x = y)) ↔ ∃x∀y([y / x]φ ↔ y = x))
14	2, 4, 13	3bitr4 158	1 ⊢ (∃!xφ ↔ ∃x(φ ∧ ∀y([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:  euex 1021  eu2 1023  kmlem15 3594

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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