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

Theorem elrabsf 3317
Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3205 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  |-  F/_ x B
Assertion
Ref Expression
elrabsf  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  [. A  /  x ]. ph ) )

Proof of Theorem elrabsf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3280 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 elrabsf.1 . . 3  |-  F/_ x B
3 nfcv 2602 . . 3  |-  F/_ y B
4 nfv 1771 . . 3  |-  F/ y
ph
5 nfsbc1v 3298 . . 3  |-  F/ x [. y  /  x ]. ph
6 sbceq1a 3289 . . 3  |-  ( x  =  y  ->  ( ph 
<-> 
[. y  /  x ]. ph ) )
72, 3, 4, 5, 6cbvrab 3054 . 2  |-  { x  e.  B  |  ph }  =  { y  e.  B  |  [. y  /  x ]. ph }
81, 7elrab2 3209 1  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  [. A  /  x ]. ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    e. wcel 1897   F/_wnfc 2589   {crab 2752   [.wsbc 3278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-rab 2757  df-v 3058  df-sbc 3279
This theorem is referenced by:  wfisg  5433  onminesb  6651  mpt2xopovel  6992  ac6num  8934  hashrabsn1  12584  bnj23  29572  bnj1204  29869  tfisg  30505  frinsg  30531  rabrenfdioph  35701
  Copyright terms: Public domain W3C validator