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Theorem rabeqf 3434
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 2917 . . 3 𝑥 𝐴 = 𝐵
4 eleq2 2825 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54anbi1d 630 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5abbid 2807 . 2 (𝐴 = 𝐵 → {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐵𝜑)})
7 df-rab 3404 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
8 df-rab 3404 . 2 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
96, 7, 83eqtr4g 2801 1 (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  {cab 2713  wnfc 2884  {crab 3403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-rab 3404
This theorem is referenced by:  fpwrelmapffs  31356  rabeq12f  36420  issmfdf  44612  smfpimltmpt  44621  smfpimltxrmptf  44633  smfpimgtmpt  44656  smfpimgtxrmptf  44659  smfsupmpt  44690  smfinfmpt  44694
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