ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elab3gf Unicode version

Theorem elab3gf 2715
Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2708. (Contributed by NM, 6-Sep-2011.)
Hypotheses
Ref Expression
elab3gf.1  |-  F/_ x A
elab3gf.2  |-  F/ x ps
elab3gf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elab3gf  |-  ( ( ps  ->  A  e.  B )  ->  ( A  e.  { x  |  ph }  <->  ps )
)

Proof of Theorem elab3gf
StepHypRef Expression
1 elab3gf.1 . . . 4  |-  F/_ x A
2 elab3gf.2 . . . 4  |-  F/ x ps
3 elab3gf.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3elabgf 2708 . . 3  |-  ( A  e.  { x  | 
ph }  ->  ( A  e.  { x  |  ph }  <->  ps )
)
54ibi 169 . 2  |-  ( A  e.  { x  | 
ph }  ->  ps )
61, 2, 3elabgf 2708 . . . 4  |-  ( A  e.  B  ->  ( A  e.  { x  |  ph }  <->  ps )
)
76imim2i 12 . . 3  |-  ( ( ps  ->  A  e.  B )  ->  ( ps  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) ) )
8 bi2 125 . . 3  |-  ( ( A  e.  { x  |  ph }  <->  ps )  ->  ( ps  ->  A  e.  { x  |  ph } ) )
97, 8syli 37 . 2  |-  ( ( ps  ->  A  e.  B )  ->  ( ps  ->  A  e.  {
x  |  ph }
) )
105, 9impbid2 135 1  |-  ( ( ps  ->  A  e.  B )  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259   F/wnf 1365    e. wcel 1409   {cab 2042   F/_wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  elab3g  2716
  Copyright terms: Public domain W3C validator