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Theorem rabbi 2536
Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2598. (Contributed by NM, 25-Nov-2013.)
Assertion
Ref Expression
rabbi (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Proof of Theorem rabbi
StepHypRef Expression
1 abbi 2196 . 2 (∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)) ↔ {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐴𝜒)})
2 df-ral 2358 . . 3 (∀𝑥𝐴 (𝜓𝜒) ↔ ∀𝑥(𝑥𝐴 → (𝜓𝜒)))
3 pm5.32 441 . . . 4 ((𝑥𝐴 → (𝜓𝜒)) ↔ ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
43albii 1400 . . 3 (∀𝑥(𝑥𝐴 → (𝜓𝜒)) ↔ ∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
52, 4bitri 182 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ ∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
6 df-rab 2362 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
7 df-rab 2362 . . 3 {𝑥𝐴𝜒} = {𝑥 ∣ (𝑥𝐴𝜒)}
86, 7eqeq12i 2096 . 2 ({𝑥𝐴𝜓} = {𝑥𝐴𝜒} ↔ {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐴𝜒)})
91, 5, 83bitr4i 210 1 (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wcel 1434  {cab 2069  wral 2353  {crab 2357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-ral 2358  df-rab 2362
This theorem is referenced by:  rabbidva  2598
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