Theorem reuhyp 5137

Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 5135. (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 1606	. 2 ⊢ ⊤
2		reuhyp.1	. . . 4 ⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶)
3	2	adantl 475	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶)
4		reuhyp.2	. . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))
5	4	3adant1 1121	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))
6	3, 5	reuhypd 5136	. 2 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
7	1, 6	mpan 680	1 ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601  ⊤wtru 1602   ∈ wcel 2107  ∃!wreu 3092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-reu 3097  df-v 3400

This theorem is referenced by:  riotaneg  11360  zriotaneg  11847  zmax  12096  rebtwnz  12098