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Theorem rabxfrd 4454
Description: Class builder membership after substituting an expression  A (containing  y) for  x in the class expression  ch. (Contributed by NM, 16-Jan-2012.)
Hypotheses
Ref Expression
rabxfrd.1  |-  F/_ y B
rabxfrd.2  |-  F/_ y C
rabxfrd.3  |-  ( (
ph  /\  y  e.  D )  ->  A  e.  D )
rabxfrd.4  |-  ( x  =  A  ->  ( ps 
<->  ch ) )
rabxfrd.5  |-  ( y  =  B  ->  A  =  C )
Assertion
Ref Expression
rabxfrd  |-  ( (
ph  /\  B  e.  D )  ->  ( C  e.  { x  e.  D  |  ps } 
<->  B  e.  { y  e.  D  |  ch } ) )
Distinct variable groups:    x, A    x, y, D    ph, y    ps, y    ch, x
Allowed substitution hints:    ph( x)    ps( x)    ch( y)    A( y)    B( x, y)    C( x, y)

Proof of Theorem rabxfrd
StepHypRef Expression
1 rabxfrd.3 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  D )  ->  A  e.  D )
21ex 114 . . . . . . . . . 10  |-  ( ph  ->  ( y  e.  D  ->  A  e.  D ) )
3 ibibr 245 . . . . . . . . . 10  |-  ( ( y  e.  D  ->  A  e.  D )  <->  ( y  e.  D  -> 
( A  e.  D  <->  y  e.  D ) ) )
42, 3sylib 121 . . . . . . . . 9  |-  ( ph  ->  ( y  e.  D  ->  ( A  e.  D  <->  y  e.  D ) ) )
54imp 123 . . . . . . . 8  |-  ( (
ph  /\  y  e.  D )  ->  ( A  e.  D  <->  y  e.  D ) )
65anbi1d 462 . . . . . . 7  |-  ( (
ph  /\  y  e.  D )  ->  (
( A  e.  D  /\  ch )  <->  ( y  e.  D  /\  ch )
) )
7 rabxfrd.4 . . . . . . . 8  |-  ( x  =  A  ->  ( ps 
<->  ch ) )
87elrab 2886 . . . . . . 7  |-  ( A  e.  { x  e.  D  |  ps }  <->  ( A  e.  D  /\  ch ) )
9 rabid 2645 . . . . . . 7  |-  ( y  e.  { y  e.  D  |  ch }  <->  ( y  e.  D  /\  ch ) )
106, 8, 93bitr4g 222 . . . . . 6  |-  ( (
ph  /\  y  e.  D )  ->  ( A  e.  { x  e.  D  |  ps } 
<->  y  e.  { y  e.  D  |  ch } ) )
1110rabbidva 2718 . . . . 5  |-  ( ph  ->  { y  e.  D  |  A  e.  { x  e.  D  |  ps } }  =  {
y  e.  D  | 
y  e.  { y  e.  D  |  ch } } )
1211eleq2d 2240 . . . 4  |-  ( ph  ->  ( B  e.  {
y  e.  D  |  A  e.  { x  e.  D  |  ps } }  <->  B  e.  { y  e.  D  |  y  e.  { y  e.  D  |  ch } } ) )
13 rabxfrd.1 . . . . 5  |-  F/_ y B
14 nfcv 2312 . . . . 5  |-  F/_ y D
15 rabxfrd.2 . . . . . 6  |-  F/_ y C
1615nfel1 2323 . . . . 5  |-  F/ y  C  e.  { x  e.  D  |  ps }
17 rabxfrd.5 . . . . . 6  |-  ( y  =  B  ->  A  =  C )
1817eleq1d 2239 . . . . 5  |-  ( y  =  B  ->  ( A  e.  { x  e.  D  |  ps } 
<->  C  e.  { x  e.  D  |  ps } ) )
1913, 14, 16, 18elrabf 2884 . . . 4  |-  ( B  e.  { y  e.  D  |  A  e. 
{ x  e.  D  |  ps } }  <->  ( B  e.  D  /\  C  e. 
{ x  e.  D  |  ps } ) )
20 nfrab1 2649 . . . . . 6  |-  F/_ y { y  e.  D  |  ch }
2113, 20nfel 2321 . . . . 5  |-  F/ y  B  e.  { y  e.  D  |  ch }
22 eleq1 2233 . . . . 5  |-  ( y  =  B  ->  (
y  e.  { y  e.  D  |  ch } 
<->  B  e.  { y  e.  D  |  ch } ) )
2313, 14, 21, 22elrabf 2884 . . . 4  |-  ( B  e.  { y  e.  D  |  y  e. 
{ y  e.  D  |  ch } }  <->  ( B  e.  D  /\  B  e. 
{ y  e.  D  |  ch } ) )
2412, 19, 233bitr3g 221 . . 3  |-  ( ph  ->  ( ( B  e.  D  /\  C  e. 
{ x  e.  D  |  ps } )  <->  ( B  e.  D  /\  B  e. 
{ y  e.  D  |  ch } ) ) )
25 pm5.32 450 . . 3  |-  ( ( B  e.  D  -> 
( C  e.  {
x  e.  D  |  ps }  <->  B  e.  { y  e.  D  |  ch } ) )  <->  ( ( B  e.  D  /\  C  e.  { x  e.  D  |  ps } )  <->  ( B  e.  D  /\  B  e. 
{ y  e.  D  |  ch } ) ) )
2624, 25sylibr 133 . 2  |-  ( ph  ->  ( B  e.  D  ->  ( C  e.  {
x  e.  D  |  ps }  <->  B  e.  { y  e.  D  |  ch } ) ) )
2726imp 123 1  |-  ( (
ph  /\  B  e.  D )  ->  ( C  e.  { x  e.  D  |  ps } 
<->  B  e.  { y  e.  D  |  ch } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   F/_wnfc 2299   {crab 2452
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732
This theorem is referenced by:  rabxfr  4455
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