Theorem reuhypd 5419

Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 7422. (Contributed by NM, 16-Jan-2012.)

Hypotheses

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

Assertion

Ref	Expression
reuhypd	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)

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

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

Proof of Theorem reuhypd

Step	Hyp	Ref	Expression
1		reuhypd.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶)
2	1	elexd 3504	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ V)
3		eueq 3714	. . . 4 ⊢ (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
4	2, 3	sylib 218	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 𝑦 = 𝐵)
5		eleq1 2829	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
6	1, 5	syl5ibrcom 247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑦 = 𝐵 → 𝑦 ∈ 𝐶))
7	6	pm4.71rd 562	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑦 = 𝐵 ↔ (𝑦 ∈ 𝐶 ∧ 𝑦 = 𝐵)))
8		reuhypd.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))
9	8	3expa 1119	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))
10	9	pm5.32da 579	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴) ↔ (𝑦 ∈ 𝐶 ∧ 𝑦 = 𝐵)))
11	7, 10	bitr4d 282	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑦 = 𝐵 ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)))
12	11	eubidv 2586	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (∃!𝑦 𝑦 = 𝐵 ↔ ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)))
13	4, 12	mpbid 232	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴))
14		df-reu 3381	. 2 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = 𝐴 ↔ ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴))
15	13, 14	sylibr 234	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∃!weu 2568  ∃!wreu 3378  Vcvv 3480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-reu 3381  df-v 3482

This theorem is referenced by:  reuhyp  5420  riotaocN  39210