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Theorem rabeqf 3458
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 2908 . . 3 𝑥 𝐴 = 𝐵
4 eleq2 2814 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54anbi1d 629 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5abbid 2795 . 2 (𝐴 = 𝐵 → {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐵𝜑)})
7 df-rab 3425 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
8 df-rab 3425 . 2 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
96, 7, 83eqtr4g 2789 1 (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  {cab 2701  wnfc 2875  {crab 3424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-rab 3425
This theorem is referenced by:  fpwrelmapffs  32454  rabeq12f  37529  issmfdf  45999  smfpimltmpt  46008  smfpimltxrmptf  46020  smfpimgtmpt  46043  smfpimgtxrmptf  46046  smfsupmpt  46077  smfinfmpt  46081
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