Theorem reuhyp 5323

Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr1 3745. (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 1541	. 2 ⊢ ⊤
2		reuhyp.1	. . . 4 ⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶)
3	2	adantl 484	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶)
4		reuhyp.2	. . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))
5	4	3adant1 1126	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))
6	3, 5	reuhypd 5322	. 2 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
7	1, 6	mpan 688	1 ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ⊤wtru 1538   ∈ wcel 2114  ∃!wreu 3142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1085  df-tru 1540  df-ex 1781  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-reu 3147  df-v 3498

This theorem is referenced by:  riotaneg  11622  zriotaneg  12099  zmax  12348  rebtwnz  12350