Theorem eqeu 3008

Description: A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)

Hypothesis

Ref	Expression
eqeu.1	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
eqeu	⊢ ((A ∈ B ∧ ψ ∧ ∀x(φ → x = A)) → ∃!xφ)

Distinct variable groups:   ψ,x   x,A

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

Proof of Theorem eqeu

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeu.1	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
2	1	spcegv 2941	. . . 4 ⊢ (A ∈ B → (ψ → ∃xφ))
3	2	imp 418	. . 3 ⊢ ((A ∈ B ∧ ψ) → ∃xφ)
4	3	3adant3 975	. 2 ⊢ ((A ∈ B ∧ ψ ∧ ∀x(φ → x = A)) → ∃xφ)
5		eqeq2 2362	. . . . . . 7 ⊢ (y = A → (x = y ↔ x = A))
6	5	imbi2d 307	. . . . . 6 ⊢ (y = A → ((φ → x = y) ↔ (φ → x = A)))
7	6	albidv 1625	. . . . 5 ⊢ (y = A → (∀x(φ → x = y) ↔ ∀x(φ → x = A)))
8	7	spcegv 2941	. . . 4 ⊢ (A ∈ B → (∀x(φ → x = A) → ∃y∀x(φ → x = y)))
9	8	imp 418	. . 3 ⊢ ((A ∈ B ∧ ∀x(φ → x = A)) → ∃y∀x(φ → x = y))
10	9	3adant2 974	. 2 ⊢ ((A ∈ B ∧ ψ ∧ ∀x(φ → x = A)) → ∃y∀x(φ → x = y))
11		nfv 1619	. . 3 ⊢ Ⅎyφ
12	11	eu3 2230	. 2 ⊢ (∃!xφ ↔ (∃xφ ∧ ∃y∀x(φ → x = y)))
13	4, 10, 12	sylanbrc 645	1 ⊢ ((A ∈ B ∧ ψ ∧ ∀x(φ → x = A)) → ∃!xφ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204

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-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by: (None)