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Theorem rabbida3 42188
 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 2824 . 2 (𝜑 → {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐵𝜒)})
4 df-rab 3079 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
5 df-rab 3079 . 2 {𝑥𝐵𝜒} = {𝑥 ∣ (𝑥𝐵𝜒)}
63, 4, 53eqtr4g 2818 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  {cab 2735  {crab 3074 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-rab 3079 This theorem is referenced by:  smflimmpt  43852  smflimsupmpt  43871  smfliminfmpt  43874
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