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Theorem rabeqf 2600
Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
Hypotheses
Ref Expression
rabeqf.1 𝑥𝐴
rabeqf.2 𝑥𝐵
Assertion
Ref Expression
rabeqf (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})

Proof of Theorem rabeqf
StepHypRef Expression
1 rabeqf.1 . . . 4 𝑥𝐴
2 rabeqf.2 . . . 4 𝑥𝐵
31, 2nfeq 2230 . . 3 𝑥 𝐴 = 𝐵
4 eleq2 2146 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54anbi1d 453 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5abbid 2199 . 2 (𝐴 = 𝐵 → {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐵𝜑)})
7 df-rab 2362 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
8 df-rab 2362 . 2 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
96, 7, 83eqtr4g 2140 1 (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  {cab 2069  wnfc 2210  {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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  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-clel 2079  df-nfc 2212  df-rab 2362
This theorem is referenced by:  rabeqif  2601  rabeq  2602
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