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Theorem rabrabi 3414
Description: Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2079 and ax-11 2093. (Revised by Gino Giotto, 20-Aug-2023.)
Hypothesis
Ref Expression
rabrabi.1 (𝑥 = 𝑦 → (𝜒𝜑))
Assertion
Ref Expression
rabrabi {𝑥 ∈ {𝑦𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜒𝜓)}
Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑥   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥)

Proof of Theorem rabrabi
StepHypRef Expression
1 rabrabi.1 . . . 4 (𝑥 = 𝑦 → (𝜒𝜑))
21cbvrabv 3412 . . 3 {𝑥𝐴𝜒} = {𝑦𝐴𝜑}
32rabeqi 3405 . 2 {𝑥 ∈ {𝑥𝐴𝜒} ∣ 𝜓} = {𝑥 ∈ {𝑦𝐴𝜑} ∣ 𝜓}
4 rabrab 3318 . 2 {𝑥 ∈ {𝑥𝐴𝜒} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜒𝜓)}
53, 4eqtr3i 2804 1 {𝑥 ∈ {𝑦𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜒𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  {crab 3092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-rab 3097
This theorem is referenced by:  wlksnwwlknvbij  27408
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