Theorem rabrabi 3417

Description: Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2139, ax-11 2156 and ax-12 2173. (Revised by Gino Giotto, 12-Oct-2024.)

Hypothesis

Ref	Expression
rabrabi.1	⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑))

Assertion

Ref	Expression
rabrabi	⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)}

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

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

Proof of Theorem rabrabi

Step	Hyp	Ref	Expression
1		df-rab 3072	. . . . . 6 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2	1	eleq2i 2830	. . . . 5 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
3		df-clab 2716	. . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ [𝑥 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜑))
4		eleq1w 2821	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
5		rabrabi.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑))
6	5	bicomd 222	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
7	6	equcoms 2024	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜒))
8	4, 7	anbi12d 630	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
9	8	sbievw 2097	. . . . . 6 ⊢ ([𝑥 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))
10	3, 9	bitri 274	. . . . 5 ⊢ (𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))
11	2, 10	bitri 274	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))
12	11	anbi1i 623	. . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜓) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜒) ∧ 𝜓))
13		anass 468	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜒) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜒 ∧ 𝜓)))
14	12, 13	bitri 274	. 2 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜒 ∧ 𝜓)))
15	14	rabbia2 3401	1 ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)}

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  [wsb 2068   ∈ wcel 2108  {cab 2715  {crab 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072

This theorem is referenced by:  wlksnwwlknvbij  28174