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Theorem rabrabi 3426
Description: Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2137, ax-11 2154 and ax-12 2171. (Revised by Gino Giotto, 12-Oct-2024.)
Hypothesis
Ref Expression
rabrabi.1 (𝑥 = 𝑦 → (𝜒𝜑))
Assertion
Ref Expression
rabrabi {𝑥 ∈ {𝑦𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜒𝜓)}
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
StepHypRef Expression
1 df-rab 3073 . . . . . 6 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
21eleq2i 2830 . . . . 5 (𝑥 ∈ {𝑦𝐴𝜑} ↔ 𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)})
3 df-clab 2716 . . . . . 6 (𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)} ↔ [𝑥 / 𝑦](𝑦𝐴𝜑))
4 eleq1w 2821 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
5 rabrabi.1 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜒𝜑))
65bicomd 222 . . . . . . . . 9 (𝑥 = 𝑦 → (𝜑𝜒))
76equcoms 2023 . . . . . . . 8 (𝑦 = 𝑥 → (𝜑𝜒))
84, 7anbi12d 631 . . . . . . 7 (𝑦 = 𝑥 → ((𝑦𝐴𝜑) ↔ (𝑥𝐴𝜒)))
98sbievw 2095 . . . . . 6 ([𝑥 / 𝑦](𝑦𝐴𝜑) ↔ (𝑥𝐴𝜒))
103, 9bitri 274 . . . . 5 (𝑥 ∈ {𝑦 ∣ (𝑦𝐴𝜑)} ↔ (𝑥𝐴𝜒))
112, 10bitri 274 . . . 4 (𝑥 ∈ {𝑦𝐴𝜑} ↔ (𝑥𝐴𝜒))
1211anbi1i 624 . . 3 ((𝑥 ∈ {𝑦𝐴𝜑} ∧ 𝜓) ↔ ((𝑥𝐴𝜒) ∧ 𝜓))
13 anass 469 . . 3 (((𝑥𝐴𝜒) ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜒𝜓)))
1412, 13bitri 274 . 2 ((𝑥 ∈ {𝑦𝐴𝜑} ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜒𝜓)))
1514rabbia2 3411 1 {𝑥 ∈ {𝑦𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜒𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  [wsb 2067  wcel 2106  {cab 2715  {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-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073
This theorem is referenced by:  wlksnwwlknvbij  28270
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