ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elabgf Unicode version

Theorem elabgf 2826
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
elabgf.1  |-  F/_ x A
elabgf.2  |-  F/ x ps
elabgf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elabgf  |-  ( A  e.  B  ->  ( A  e.  { x  |  ph }  <->  ps )
)

Proof of Theorem elabgf
StepHypRef Expression
1 elabgf.1 . 2  |-  F/_ x A
2 nfab1 2283 . . . 4  |-  F/_ x { x  |  ph }
31, 2nfel 2290 . . 3  |-  F/ x  A  e.  { x  |  ph }
4 elabgf.2 . . 3  |-  F/ x ps
53, 4nfbi 1568 . 2  |-  F/ x
( A  e.  {
x  |  ph }  <->  ps )
6 eleq1 2202 . . 3  |-  ( x  =  A  ->  (
x  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
7 elabgf.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
86, 7bibi12d 234 . 2  |-  ( x  =  A  ->  (
( x  e.  {
x  |  ph }  <->  ph )  <->  ( A  e. 
{ x  |  ph } 
<->  ps ) ) )
9 abid 2127 . 2  |-  ( x  e.  { x  | 
ph }  <->  ph )
101, 5, 8, 9vtoclgf 2744 1  |-  ( A  e.  B  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331   F/wnf 1436    e. wcel 1480   {cab 2125   F/_wnfc 2268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688
This theorem is referenced by:  elabf  2827  elabg  2830  elab3gf  2834  elrabf  2838  bj-intabssel  13080
  Copyright terms: Public domain W3C validator