Theorem reuhyp 4563

Description: A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.)

Hypotheses

Ref	Expression
reuhyp.1	⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶)
reuhyp.2	⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))

Assertion

Ref	Expression
reuhyp	⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)

Distinct variable groups:   𝑦,𝐵   𝑦,𝐶   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem reuhyp

Step	Hyp	Ref	Expression
1		tru 1399	. 2 ⊢ ⊤
2		reuhyp.1	. . . 4 ⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶)
3	2	adantl 277	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶)
4		reuhyp.2	. . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))
5	4	3adant1 1039	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))
6	3, 5	reuhypd 4562	. 2 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
7	1, 6	mpan 424	1 ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ⊤wtru 1396   ∈ wcel 2200  ∃!wreu 2510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-reu 2515  df-v 2801

This theorem is referenced by: (None)