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Theorem rabbi 2481
 Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2542. (Contributed by NM, 25-Nov-2013.)
Assertion
Ref Expression
rabbi (x A (ψχ) ↔ {x Aψ} = {x Aχ})

Proof of Theorem rabbi
StepHypRef Expression
1 abbi 2148 . 2 (x((x A ψ) ↔ (x A χ)) ↔ {x ∣ (x A ψ)} = {x ∣ (x A χ)})
2 df-ral 2305 . . 3 (x A (ψχ) ↔ x(x A → (ψχ)))
3 pm5.32 426 . . . 4 ((x A → (ψχ)) ↔ ((x A ψ) ↔ (x A χ)))
43albii 1356 . . 3 (x(x A → (ψχ)) ↔ x((x A ψ) ↔ (x A χ)))
52, 4bitri 173 . 2 (x A (ψχ) ↔ x((x A ψ) ↔ (x A χ)))
6 df-rab 2309 . . 3 {x Aψ} = {x ∣ (x A ψ)}
7 df-rab 2309 . . 3 {x Aχ} = {x ∣ (x A χ)}
86, 7eqeq12i 2050 . 2 ({x Aψ} = {x Aχ} ↔ {x ∣ (x A ψ)} = {x ∣ (x A χ)})
91, 5, 83bitr4i 201 1 (x A (ψχ) ↔ {x Aψ} = {x Aχ})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  {crab 2304 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-ral 2305  df-rab 2309 This theorem is referenced by:  rabbidva  2542
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