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Theorem rabbida2 41275
Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
rabbida2.1 𝑥𝜑
rabbida2.2 (𝜑𝐴 = 𝐵)
rabbida2.3 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
rabbida2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Proof of Theorem rabbida2
StepHypRef Expression
1 rabbida2.1 . . 3 𝑥𝜑
2 rabbida2.2 . . . . 5 (𝜑𝐴 = 𝐵)
32eleq2d 2895 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐵))
4 rabbida2.3 . . . 4 (𝜑 → (𝜓𝜒))
53, 4anbi12d 630 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
61, 5abbid 2884 . 2 (𝜑 → {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐵𝜒)})
7 df-rab 3144 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
8 df-rab 3144 . 2 {𝑥𝐵𝜒} = {𝑥 ∣ (𝑥𝐵𝜒)}
96, 7, 83eqtr4g 2878 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wnf 1775  wcel 2105  {cab 2796  {crab 3139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-rab 3144
This theorem is referenced by:  smflimmpt  42961
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