Theorem riotarab 7411

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

Hypothesis

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

Assertion

Ref	Expression
riotarab	⊢ (℩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 riotarab

Step	Hyp	Ref	Expression
1		riotarab.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	bicomd 226	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑))
3	2	equcoms 2049	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
4	3	elrab 3649	. . . . 5 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
5	4	anbi1i 635	. . . 4 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ∧ 𝜒) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜒))
6		anass 473	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒)))
7	5, 6	bitri 278	. . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒)))
8	7	iotabii 6521	. 2 ⊢ (℩𝑥(𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ∧ 𝜒)) = (℩𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒)))
9		df-riota 7369	. 2 ⊢ (℩𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓}𝜒) = (℩𝑥(𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓} ∧ 𝜒))
10		df-riota 7369	. 2 ⊢ (℩𝑥 ∈ 𝐴 (𝜑 ∧ 𝜒)) = (℩𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒)))
11	8, 9, 10	3eqtr4i 2795	1 ⊢ (℩𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓}𝜒) = (℩𝑥 ∈ 𝐴 (𝜑 ∧ 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {crab 3415  ℩cio 6490  ℩crio 7368

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-ss 3921  df-uni 4872  df-iota 6492  df-riota 7369

This theorem is used by:  eqcuts  27989