MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabeqf Structured version   Visualization version   GIF version

Theorem rabeqf 3470
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 2828 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54anbi1d 631 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5abbid 2808 . 2 (𝐴 = 𝐵 → {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐵𝜑)})
7 df-rab 3434 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
8 df-rab 3434 . 2 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
96, 7, 83eqtr4g 2800 1 (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wnfc 2888  {crab 3433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434
This theorem is referenced by:  fpwrelmapffs  32752  rabeq12f  38144  issmfdf  46693  smfpimltmpt  46702  smfpimltxrmptf  46714  smfpimgtmpt  46737  smfpimgtxrmptf  46740  smfsupmpt  46771
  Copyright terms: Public domain W3C validator