Theorem eqeu 2762

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 2686	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝜓 → ∃𝑥𝜑))
3	2	imp 122	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥𝜑)
4	3	3adant3 958	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃𝑥𝜑)
5		eqeq2 2090	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
6	5	imbi2d 228	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝐴)))
7	6	albidv 1745	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝐴)))
8	7	spcegv 2686	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
9	8	imp 122	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
10	9	3adant2 957	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
11		nfv 1461	. . 3 ⊢ Ⅎ𝑦𝜑
12	11	eu3 1987	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
13	4, 10, 12	sylanbrc 408	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃!𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 103   ∧ w3a 919  ∀wal 1282   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∃!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by: (None)