Theorem reuhypd 5058

Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 6834. (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 3367	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ V)
3		eueq 3536	. . . 4 ⊢ (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
4	2, 3	sylib 209	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 𝑦 = 𝐵)
5		eleq1 2832	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
6	1, 5	syl5ibrcom 238	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑦 = 𝐵 → 𝑦 ∈ 𝐶))
7	6	pm4.71rd 558	. . . . 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 1147	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))
10	9	pm5.32da 574	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴) ↔ (𝑦 ∈ 𝐶 ∧ 𝑦 = 𝐵)))
11	7, 10	bitr4d 273	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑦 = 𝐵 ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)))
12	11	eubidv 2585	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (∃!𝑦 𝑦 = 𝐵 ↔ ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)))
13	4, 12	mpbid 223	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴))
14		df-reu 3062	. 2 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = 𝐴 ↔ ∃!𝑦(𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴))
15	13, 14	sylibr 225	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155  ∃!weu 2581  ∃!wreu 3057  Vcvv 3350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-reu 3062  df-v 3352

This theorem is referenced by:  reuhyp  5059  riotaocN  35097