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Theorem cbvrabv2w 45016
Description: A more general version of cbvrabv 3443. Version of cbvrabv2 45015 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Revised by GG, 14-Aug-2025.)
Hypotheses
Ref Expression
cbvrabv2w.1 (𝑥 = 𝑦𝐴 = 𝐵)
cbvrabv2w.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrabv2w {𝑥𝐴𝜑} = {𝑦𝐵𝜓}
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvrabv2w
StepHypRef Expression
1 id 22 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
2 cbvrabv2w.1 . . . . 5 (𝑥 = 𝑦𝐴 = 𝐵)
31, 2eleq12d 2831 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐵))
4 cbvrabv2w.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4anbi12d 631 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐵𝜓)))
65cbvabv 2808 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ (𝑦𝐵𝜓)}
7 df-rab 3433 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
8 df-rab 3433 . 2 {𝑦𝐵𝜓} = {𝑦 ∣ (𝑦𝐵𝜓)}
96, 7, 83eqtr4i 2771 1 {𝑥𝐴𝜑} = {𝑦𝐵𝜓}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1535  wcel 2104  {cab 2710  {crab 3432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-rab 3433
This theorem is referenced by:  smfsuplem2  46718
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