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Theorem rabbia2 3218
Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
rabbia2.1 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
Assertion
Ref Expression
rabbia2 {𝑥𝐴𝜓} = {𝑥𝐵𝜒}

Proof of Theorem rabbia2
StepHypRef Expression
1 rabbia2.1 . . . 4 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
21a1i 11 . . 3 (⊤ → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
32rabbidva2 3217 . 2 (⊤ → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
43trud 1533 1 {𝑥𝐴𝜓} = {𝑥𝐵𝜒}
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wtru 1524  wcel 2030  {crab 2945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-rab 2950
This theorem is referenced by:  finsumvtxdg2ssteplem3  26499  clwwlknon2x  27078  numclwwlkovh  27353  smflim  41306  smflim2  41333  smflimsuplem1  41347  smflimsup  41355  sprvalpwn0  42058
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