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

Theorem elrabsf 3834
Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3688 has implicit substitution). The hypothesis specifies that 𝑥 must not be a free variable in 𝐵. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Hypothesis
Ref Expression
elrabsf.1 𝑥𝐵
Assertion
Ref Expression
elrabsf (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵[𝐴 / 𝑥]𝜑))

Proof of Theorem elrabsf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3790 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 elrabsf.1 . . 3 𝑥𝐵
3 nfcv 2905 . . 3 𝑦𝐵
4 nfv 1914 . . 3 𝑦𝜑
5 nfsbc1v 3808 . . 3 𝑥[𝑦 / 𝑥]𝜑
6 sbceq1a 3799 . . 3 (𝑥 = 𝑦 → (𝜑[𝑦 / 𝑥]𝜑))
72, 3, 4, 5, 6cbvrabw 3473 . 2 {𝑥𝐵𝜑} = {𝑦𝐵[𝑦 / 𝑥]𝜑}
81, 7elrab2 3695 1 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵[𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2108  wnfc 2890  {crab 3436  [wsbc 3788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-sbc 3789
This theorem is referenced by:  frpoinsg  6364  wfisgOLD  6375  onminesb  7813  tfisg  7875  mpoxopovel  8245  frinsg  9791  ac6num  10519  hashrabsn1  14413  bnj23  34732  bnj1204  35026  weiunlem2  36464  rabrenfdioph  42825
  Copyright terms: Public domain W3C validator