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Theorem rabbi 2487
 Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2548. (Contributed by NM, 25-Nov-2013.)
Assertion
Ref Expression
rabbi (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Proof of Theorem rabbi
StepHypRef Expression
1 abbi 2151 . 2 (∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)) ↔ {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐴𝜒)})
2 df-ral 2311 . . 3 (∀𝑥𝐴 (𝜓𝜒) ↔ ∀𝑥(𝑥𝐴 → (𝜓𝜒)))
3 pm5.32 426 . . . 4 ((𝑥𝐴 → (𝜓𝜒)) ↔ ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
43albii 1359 . . 3 (∀𝑥(𝑥𝐴 → (𝜓𝜒)) ↔ ∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
52, 4bitri 173 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ ∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
6 df-rab 2315 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
7 df-rab 2315 . . 3 {𝑥𝐴𝜒} = {𝑥 ∣ (𝑥𝐴𝜒)}
86, 7eqeq12i 2053 . 2 ({𝑥𝐴𝜓} = {𝑥𝐴𝜒} ↔ {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐴𝜒)})
91, 5, 83bitr4i 201 1 (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  {crab 2310 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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-ral 2311  df-rab 2315 This theorem is referenced by:  rabbidva  2548
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