Theorem rmo3 3134

Description: Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

Hypothesis

Ref	Expression
rmo2.1	⊢ Ⅎyφ

Assertion

Ref	Expression
rmo3	⊢ (∃*x ∈ A φ ↔ ∀x ∈ A ∀y ∈ A ((φ ∧ [y / x]φ) → x = y))

Distinct variable group:   x,y,A

Allowed substitution hints:   φ(x,y)

Proof of Theorem rmo3

Step	Hyp	Ref	Expression
1		df-rmo 2623	. 2 ⊢ (∃*x ∈ A φ ↔ ∃*x(x ∈ A ∧ φ))
2		sban 2069	. . . . . . . . . . 11 ⊢ ([y / x](x ∈ A ∧ φ) ↔ ([y / x]x ∈ A ∧ [y / x]φ))
3		clelsb1 2455	. . . . . . . . . . . 12 ⊢ ([y / x]x ∈ A ↔ y ∈ A)
4	3	anbi1i 676	. . . . . . . . . . 11 ⊢ (([y / x]x ∈ A ∧ [y / x]φ) ↔ (y ∈ A ∧ [y / x]φ))
5	2, 4	bitri 240	. . . . . . . . . 10 ⊢ ([y / x](x ∈ A ∧ φ) ↔ (y ∈ A ∧ [y / x]φ))
6	5	anbi2i 675	. . . . . . . . 9 ⊢ (((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) ↔ ((x ∈ A ∧ φ) ∧ (y ∈ A ∧ [y / x]φ)))
7		an4 797	. . . . . . . . 9 ⊢ (((x ∈ A ∧ φ) ∧ (y ∈ A ∧ [y / x]φ)) ↔ ((x ∈ A ∧ y ∈ A) ∧ (φ ∧ [y / x]φ)))
8		ancom 437	. . . . . . . . . 10 ⊢ ((x ∈ A ∧ y ∈ A) ↔ (y ∈ A ∧ x ∈ A))
9	8	anbi1i 676	. . . . . . . . 9 ⊢ (((x ∈ A ∧ y ∈ A) ∧ (φ ∧ [y / x]φ)) ↔ ((y ∈ A ∧ x ∈ A) ∧ (φ ∧ [y / x]φ)))
10	6, 7, 9	3bitri 262	. . . . . . . 8 ⊢ (((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) ↔ ((y ∈ A ∧ x ∈ A) ∧ (φ ∧ [y / x]φ)))
11	10	imbi1i 315	. . . . . . 7 ⊢ ((((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y) ↔ (((y ∈ A ∧ x ∈ A) ∧ (φ ∧ [y / x]φ)) → x = y))
12		impexp 433	. . . . . . 7 ⊢ ((((y ∈ A ∧ x ∈ A) ∧ (φ ∧ [y / x]φ)) → x = y) ↔ ((y ∈ A ∧ x ∈ A) → ((φ ∧ [y / x]φ) → x = y)))
13		impexp 433	. . . . . . 7 ⊢ (((y ∈ A ∧ x ∈ A) → ((φ ∧ [y / x]φ) → x = y)) ↔ (y ∈ A → (x ∈ A → ((φ ∧ [y / x]φ) → x = y))))
14	11, 12, 13	3bitri 262	. . . . . 6 ⊢ ((((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y) ↔ (y ∈ A → (x ∈ A → ((φ ∧ [y / x]φ) → x = y))))
15	14	albii 1566	. . . . 5 ⊢ (∀y(((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y) ↔ ∀y(y ∈ A → (x ∈ A → ((φ ∧ [y / x]φ) → x = y))))
16		df-ral 2620	. . . . 5 ⊢ (∀y ∈ A (x ∈ A → ((φ ∧ [y / x]φ) → x = y)) ↔ ∀y(y ∈ A → (x ∈ A → ((φ ∧ [y / x]φ) → x = y))))
17		r19.21v 2702	. . . . 5 ⊢ (∀y ∈ A (x ∈ A → ((φ ∧ [y / x]φ) → x = y)) ↔ (x ∈ A → ∀y ∈ A ((φ ∧ [y / x]φ) → x = y)))
18	15, 16, 17	3bitr2i 264	. . . 4 ⊢ (∀y(((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y) ↔ (x ∈ A → ∀y ∈ A ((φ ∧ [y / x]φ) → x = y)))
19	18	albii 1566	. . 3 ⊢ (∀x∀y(((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y) ↔ ∀x(x ∈ A → ∀y ∈ A ((φ ∧ [y / x]φ) → x = y)))
20		nfv 1619	. . . . 5 ⊢ Ⅎy x ∈ A
21		rmo2.1	. . . . 5 ⊢ Ⅎyφ
22	20, 21	nfan 1824	. . . 4 ⊢ Ⅎy(x ∈ A ∧ φ)
23	22	mo3 2235	. . 3 ⊢ (∃*x(x ∈ A ∧ φ) ↔ ∀x∀y(((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y))
24		df-ral 2620	. . 3 ⊢ (∀x ∈ A ∀y ∈ A ((φ ∧ [y / x]φ) → x = y) ↔ ∀x(x ∈ A → ∀y ∈ A ((φ ∧ [y / x]φ) → x = y)))
25	19, 23, 24	3bitr4i 268	. 2 ⊢ (∃*x(x ∈ A ∧ φ) ↔ ∀x ∈ A ∀y ∈ A ((φ ∧ [y / x]φ) → x = y))
26	1, 25	bitri 240	1 ⊢ (∃*x ∈ A φ ↔ ∀x ∈ A ∀y ∈ A ((φ ∧ [y / x]φ) → x = y))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544  [wsb 1648   ∈ wcel 1710  ∃*wmo 2205  ∀wral 2615  ∃*wrmo 2618

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  ax-ext 2334

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  df-mo 2209  df-cleq 2346  df-clel 2349  df-ral 2620  df-rmo 2623

This theorem is referenced by: (None)