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Theorem rabbi 2580
Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2643. (Contributed by NM, 25-Nov-2013.)
Assertion
Ref Expression
rabbi (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Proof of Theorem rabbi
StepHypRef Expression
1 abbi 2226 . 2 (∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)) ↔ {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐴𝜒)})
2 df-ral 2393 . . 3 (∀𝑥𝐴 (𝜓𝜒) ↔ ∀𝑥(𝑥𝐴 → (𝜓𝜒)))
3 pm5.32 446 . . . 4 ((𝑥𝐴 → (𝜓𝜒)) ↔ ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
43albii 1427 . . 3 (∀𝑥(𝑥𝐴 → (𝜓𝜒)) ↔ ∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
52, 4bitri 183 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ ∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
6 df-rab 2397 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
7 df-rab 2397 . . 3 {𝑥𝐴𝜒} = {𝑥 ∣ (𝑥𝐴𝜒)}
86, 7eqeq12i 2126 . 2 ({𝑥𝐴𝜓} = {𝑥𝐴𝜒} ↔ {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐴𝜒)})
91, 5, 83bitr4i 211 1 (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1310   = wceq 1312  wcel 1461  {cab 2099  wral 2388  {crab 2392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-11 1465  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-ral 2393  df-rab 2397
This theorem is referenced by:  rabbidva  2643
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