Theorem reupick2 3542

Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Assertion

Ref	Expression
reupick2	⊢ (((∀x ∈ A (ψ → φ) ∧ ∃x ∈ A ψ ∧ ∃!x ∈ A φ) ∧ x ∈ A) → (φ ↔ ψ))

Distinct variable group:   x,A

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

Proof of Theorem reupick2

Step	Hyp	Ref	Expression
1		ancr 532	. . . . . 6 ⊢ ((ψ → φ) → (ψ → (φ ∧ ψ)))
2	1	ralimi 2690	. . . . 5 ⊢ (∀x ∈ A (ψ → φ) → ∀x ∈ A (ψ → (φ ∧ ψ)))
3		rexim 2719	. . . . 5 ⊢ (∀x ∈ A (ψ → (φ ∧ ψ)) → (∃x ∈ A ψ → ∃x ∈ A (φ ∧ ψ)))
4	2, 3	syl 15	. . . 4 ⊢ (∀x ∈ A (ψ → φ) → (∃x ∈ A ψ → ∃x ∈ A (φ ∧ ψ)))
5		reupick3 3541	. . . . . 6 ⊢ ((∃!x ∈ A φ ∧ ∃x ∈ A (φ ∧ ψ) ∧ x ∈ A) → (φ → ψ))
6	5	3exp 1150	. . . . 5 ⊢ (∃!x ∈ A φ → (∃x ∈ A (φ ∧ ψ) → (x ∈ A → (φ → ψ))))
7	6	com12 27	. . . 4 ⊢ (∃x ∈ A (φ ∧ ψ) → (∃!x ∈ A φ → (x ∈ A → (φ → ψ))))
8	4, 7	syl6 29	. . 3 ⊢ (∀x ∈ A (ψ → φ) → (∃x ∈ A ψ → (∃!x ∈ A φ → (x ∈ A → (φ → ψ)))))
9	8	3imp1 1164	. 2 ⊢ (((∀x ∈ A (ψ → φ) ∧ ∃x ∈ A ψ ∧ ∃!x ∈ A φ) ∧ x ∈ A) → (φ → ψ))
10		rsp 2675	. . . 4 ⊢ (∀x ∈ A (ψ → φ) → (x ∈ A → (ψ → φ)))
11	10	3ad2ant1 976	. . 3 ⊢ ((∀x ∈ A (ψ → φ) ∧ ∃x ∈ A ψ ∧ ∃!x ∈ A φ) → (x ∈ A → (ψ → φ)))
12	11	imp 418	. 2 ⊢ (((∀x ∈ A (ψ → φ) ∧ ∃x ∈ A ψ ∧ ∃!x ∈ A φ) ∧ x ∈ A) → (ψ → φ))
13	9, 12	impbid 183	1 ⊢ (((∀x ∈ A (ψ → φ) ∧ ∃x ∈ A ψ ∧ ∃!x ∈ A φ) ∧ x ∈ A) → (φ ↔ ψ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  ∃!wreu 2617

This proof depends on 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 proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-ral 2620  df-rex 2621  df-reu 2622

This theorem is used by: (None)