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Theorem rabbida3 44278
Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
rabbida3.1 𝑥𝜑
rabbida3.2 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
Assertion
Ref Expression
rabbida3 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Proof of Theorem rabbida3
StepHypRef Expression
1 rabbida3.1 . . 3 𝑥𝜑
2 rabbida3.2 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
31, 2abbid 2795 . 2 (𝜑 → {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐵𝜒)})
4 df-rab 3425 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
5 df-rab 3425 . 2 {𝑥𝐵𝜒} = {𝑥 ∣ (𝑥𝐵𝜒)}
63, 4, 53eqtr4g 2789 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wnf 1777  wcel 2098  {cab 2701  {crab 3424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-rab 3425
This theorem is referenced by:  smflimmpt  45977  smflimsupmpt  45996  smfliminfmpt  45999
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