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Theorem rabeqbidva 3450
Description: Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.) Remove DV conditions. (Revised by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
rabeqbidva.1 (𝜑𝐴 = 𝐵)
rabeqbidva.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rabeqbidva (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem rabeqbidva
StepHypRef Expression
1 rabeqbidva.2 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
21rabbidva 3440 . 2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
3 rabeqbidva.1 . . . . 5 (𝜑𝐴 = 𝐵)
43eleq2d 2825 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐵))
54anbi1d 631 . . 3 (𝜑 → ((𝑥𝐴𝜒) ↔ (𝑥𝐵𝜒)))
65rabbidva2 3435 . 2 (𝜑 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})
72, 6eqtrd 2775 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {crab 3433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434
This theorem is referenced by:  rabeqbidv  3452  natpropd  18033  gsumpropd2lem  18705  elntg  29014  rmfsupp2  33228  poimirlem28  37635  scotteqd  44233  uspgrlimlem1  47891  domnmsuppn0  48214  eenglngeehlnm  48589
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