ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elrab3t Unicode version

Theorem elrab3t 2877
Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2879.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
Assertion
Ref Expression
elrab3t  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( A  e.  {
x  e.  B  |  ph }  <->  ps ) )
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem elrab3t
StepHypRef Expression
1 simpr 109 . . 3  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  A  e.  B )
2 nfa1 1528 . . . . 5  |-  F/ x A. x ( x  =  A  ->  ( ph  <->  ps ) )
3 nfv 1515 . . . . 5  |-  F/ x  A  e.  B
42, 3nfan 1552 . . . 4  |-  F/ x
( A. x ( x  =  A  -> 
( ph  <->  ps ) )  /\  A  e.  B )
5 simpl 108 . . . . . 6  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  A. x ( x  =  A  ->  ( ph 
<->  ps ) ) )
6519.21bi 1545 . . . . 5  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( x  =  A  ->  ( ph  <->  ps )
) )
7 eleq1 2227 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
87biimparc 297 . . . . . . . . 9  |-  ( ( A  e.  B  /\  x  =  A )  ->  x  e.  B )
98biantrurd 303 . . . . . . . 8  |-  ( ( A  e.  B  /\  x  =  A )  ->  ( ph  <->  ( x  e.  B  /\  ph )
) )
109bibi1d 232 . . . . . . 7  |-  ( ( A  e.  B  /\  x  =  A )  ->  ( ( ph  <->  ps )  <->  ( ( x  e.  B  /\  ph )  <->  ps )
) )
1110pm5.74da 440 . . . . . 6  |-  ( A  e.  B  ->  (
( x  =  A  ->  ( ph  <->  ps )
)  <->  ( x  =  A  ->  ( (
x  e.  B  /\  ph )  <->  ps ) ) ) )
1211adantl 275 . . . . 5  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( ( x  =  A  ->  ( ph  <->  ps ) )  <->  ( x  =  A  ->  ( ( x  e.  B  /\  ph )  <->  ps ) ) ) )
136, 12mpbid 146 . . . 4  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( x  =  A  ->  ( ( x  e.  B  /\  ph ) 
<->  ps ) ) )
144, 13alrimi 1509 . . 3  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  A. x ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ps )
) )
15 elabgt 2863 . . 3  |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  ( (
x  e.  B  /\  ph )  <->  ps ) ) )  ->  ( A  e. 
{ x  |  ( x  e.  B  /\  ph ) }  <->  ps )
)
161, 14, 15syl2anc 409 . 2  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( A  e.  {
x  |  ( x  e.  B  /\  ph ) }  <->  ps ) )
17 df-rab 2451 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
1817eleq2i 2231 . . 3  |-  ( A  e.  { x  e.  B  |  ph }  <->  A  e.  { x  |  ( x  e.  B  /\  ph ) } )
1918bibi1i 227 . 2  |-  ( ( A  e.  { x  e.  B  |  ph }  <->  ps )  <->  ( A  e. 
{ x  |  ( x  e.  B  /\  ph ) }  <->  ps )
)
2016, 19sylibr 133 1  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( A  e.  {
x  e.  B  |  ph }  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1340    = wceq 1342    e. wcel 2135   {cab 2150   {crab 2446
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rab 2451  df-v 2724
This theorem is referenced by:  f1oresrab  5645
  Copyright terms: Public domain W3C validator