Theorem reurab 33674

Description: Restricted existential uniqueness of a restricted abstraction. (Contributed by Scott Fenton, 8-Aug-2024.)

Hypothesis

Ref	Expression
reurab.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
reurab	⊢ (∃!𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓}𝜒 ↔ ∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜒))

Distinct variable groups:   𝑦,𝐴   𝜑,𝑦   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem reurab

Step	Hyp	Ref	Expression
1		reurab.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	bicomd 222	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑))
3	2	equcoms 2023	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
4	3	elrab 3624	. . . . 5 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
5	4	anbi1i 624	. . . 4 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ∧ 𝜒) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜒))
6		anass 469	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒)))
7	5, 6	bitri 274	. . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒)))
8	7	eubii 2585	. 2 ⊢ (∃!𝑥(𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ∧ 𝜒) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒)))
9		df-reu 3072	. 2 ⊢ (∃!𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓}𝜒 ↔ ∃!𝑥(𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ∧ 𝜒))
10		df-reu 3072	. 2 ⊢ (∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜒) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒)))
11	8, 9, 10	3bitr4i 303	1 ⊢ (∃!𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓}𝜒 ↔ ∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  ∃!weu 2568  ∃!wreu 3066  {crab 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-reu 3072  df-rab 3073  df-v 3434

This theorem is referenced by:  eqscut  33999