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Theorem rabrabiOLD 3404
Description: Obsolete version of rabrabi 3403 as of 12-Oct-2024. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2141 and ax-11 2158. (Revised by Gino Giotto, 20-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
rabrabiOLD.1 (𝑥 = 𝑦 → (𝜒𝜑))
Assertion
Ref Expression
rabrabiOLD {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 rabrabiOLD
StepHypRef Expression
1 rabrabiOLD.1 . . . 4 (𝑥 = 𝑦 → (𝜒𝜑))
21cbvrabv 3402 . . 3 {𝑥𝐴𝜒} = {𝑦𝐴𝜑}
32rabeqi 3392 . 2 {𝑥 ∈ {𝑥𝐴𝜒} ∣ 𝜓} = {𝑥 ∈ {𝑦𝐴𝜑} ∣ 𝜓}
4 rabrab 3291 . 2 {𝑥 ∈ {𝑥𝐴𝜒} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜒𝜓)}
53, 4eqtr3i 2767 1 {𝑥 ∈ {𝑦𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜒𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  {crab 3065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070
This theorem is referenced by: (None)
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