Theorem rmo4 3030

Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.)

Hypothesis

Ref	Expression
rmo4.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
rmo4	⊢ (∃*x ∈ A φ ↔ ∀x ∈ A ∀y ∈ A ((φ ∧ ψ) → x = y))

Distinct variable groups:   x,y,A   φ,y   ψ,x

Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem rmo4

Step	Hyp	Ref	Expression
1		df-rmo 2623	. 2 ⊢ (∃*x ∈ A φ ↔ ∃*x(x ∈ A ∧ φ))
2		an4 797	. . . . . . . . 9 ⊢ (((x ∈ A ∧ φ) ∧ (y ∈ A ∧ ψ)) ↔ ((x ∈ A ∧ y ∈ A) ∧ (φ ∧ ψ)))
3		ancom 437	. . . . . . . . . 10 ⊢ ((x ∈ A ∧ y ∈ A) ↔ (y ∈ A ∧ x ∈ A))
4	3	anbi1i 676	. . . . . . . . 9 ⊢ (((x ∈ A ∧ y ∈ A) ∧ (φ ∧ ψ)) ↔ ((y ∈ A ∧ x ∈ A) ∧ (φ ∧ ψ)))
5	2, 4	bitri 240	. . . . . . . 8 ⊢ (((x ∈ A ∧ φ) ∧ (y ∈ A ∧ ψ)) ↔ ((y ∈ A ∧ x ∈ A) ∧ (φ ∧ ψ)))
6	5	imbi1i 315	. . . . . . 7 ⊢ ((((x ∈ A ∧ φ) ∧ (y ∈ A ∧ ψ)) → x = y) ↔ (((y ∈ A ∧ x ∈ A) ∧ (φ ∧ ψ)) → x = y))
7		impexp 433	. . . . . . 7 ⊢ ((((y ∈ A ∧ x ∈ A) ∧ (φ ∧ ψ)) → x = y) ↔ ((y ∈ A ∧ x ∈ A) → ((φ ∧ ψ) → x = y)))
8		impexp 433	. . . . . . 7 ⊢ (((y ∈ A ∧ x ∈ A) → ((φ ∧ ψ) → x = y)) ↔ (y ∈ A → (x ∈ A → ((φ ∧ ψ) → x = y))))
9	6, 7, 8	3bitri 262	. . . . . 6 ⊢ ((((x ∈ A ∧ φ) ∧ (y ∈ A ∧ ψ)) → x = y) ↔ (y ∈ A → (x ∈ A → ((φ ∧ ψ) → x = y))))
10	9	albii 1566	. . . . 5 ⊢ (∀y(((x ∈ A ∧ φ) ∧ (y ∈ A ∧ ψ)) → x = y) ↔ ∀y(y ∈ A → (x ∈ A → ((φ ∧ ψ) → x = y))))
11		df-ral 2620	. . . . 5 ⊢ (∀y ∈ A (x ∈ A → ((φ ∧ ψ) → x = y)) ↔ ∀y(y ∈ A → (x ∈ A → ((φ ∧ ψ) → x = y))))
12		r19.21v 2702	. . . . 5 ⊢ (∀y ∈ A (x ∈ A → ((φ ∧ ψ) → x = y)) ↔ (x ∈ A → ∀y ∈ A ((φ ∧ ψ) → x = y)))
13	10, 11, 12	3bitr2i 264	. . . 4 ⊢ (∀y(((x ∈ A ∧ φ) ∧ (y ∈ A ∧ ψ)) → x = y) ↔ (x ∈ A → ∀y ∈ A ((φ ∧ ψ) → x = y)))
14	13	albii 1566	. . 3 ⊢ (∀x∀y(((x ∈ A ∧ φ) ∧ (y ∈ A ∧ ψ)) → x = y) ↔ ∀x(x ∈ A → ∀y ∈ A ((φ ∧ ψ) → x = y)))
15		eleq1 2413	. . . . 5 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
16		rmo4.1	. . . . 5 ⊢ (x = y → (φ ↔ ψ))
17	15, 16	anbi12d 691	. . . 4 ⊢ (x = y → ((x ∈ A ∧ φ) ↔ (y ∈ A ∧ ψ)))
18	17	mo4 2237	. . 3 ⊢ (∃*x(x ∈ A ∧ φ) ↔ ∀x∀y(((x ∈ A ∧ φ) ∧ (y ∈ A ∧ ψ)) → x = y))
19		df-ral 2620	. . 3 ⊢ (∀x ∈ A ∀y ∈ A ((φ ∧ ψ) → x = y) ↔ ∀x(x ∈ A → ∀y ∈ A ((φ ∧ ψ) → x = y)))
20	14, 18, 19	3bitr4i 268	. 2 ⊢ (∃*x(x ∈ A ∧ φ) ↔ ∀x ∈ A ∀y ∈ A ((φ ∧ ψ) → x = y))
21	1, 20	bitri 240	1 ⊢ (∃*x ∈ A φ ↔ ∀x ∈ A ∀y ∈ A ((φ ∧ ψ) → x = y))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ 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:  reu4  3031