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Theorem eqerlem 6565
Description: Lemma for eqer 6566. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
Hypotheses
Ref Expression
eqer.1  |-  ( x  =  y  ->  A  =  B )
eqer.2  |-  R  =  { <. x ,  y
>.  |  A  =  B }
Assertion
Ref Expression
eqerlem  |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
)
Distinct variable groups:    x, w, y   
x, z, y    y, A    x, B
Allowed substitution hints:    A( x, z, w)    B( y, z, w)    R( x, y, z, w)

Proof of Theorem eqerlem
StepHypRef Expression
1 eqer.2 . . 3  |-  R  =  { <. x ,  y
>.  |  A  =  B }
21brabsb 4261 . 2  |-  ( z R w  <->  [. z  /  x ]. [. w  / 
y ]. A  =  B )
3 vex 2740 . . 3  |-  z  e. 
_V
4 nfcsb1v 3090 . . . . 5  |-  F/_ x [_ z  /  x ]_ A
5 nfcsb1v 3090 . . . . 5  |-  F/_ x [_ w  /  x ]_ A
64, 5nfeq 2327 . . . 4  |-  F/ x [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
7 vex 2740 . . . . . 6  |-  w  e. 
_V
8 nfv 1528 . . . . . . 7  |-  F/ y  A  =  [_ w  /  x ]_ A
9 vex 2740 . . . . . . . . . 10  |-  y  e. 
_V
10 nfcv 2319 . . . . . . . . . 10  |-  F/_ x B
11 eqer.1 . . . . . . . . . 10  |-  ( x  =  y  ->  A  =  B )
129, 10, 11csbief 3101 . . . . . . . . 9  |-  [_ y  /  x ]_ A  =  B
13 csbeq1 3060 . . . . . . . . 9  |-  ( y  =  w  ->  [_ y  /  x ]_ A  = 
[_ w  /  x ]_ A )
1412, 13eqtr3id 2224 . . . . . . . 8  |-  ( y  =  w  ->  B  =  [_ w  /  x ]_ A )
1514eqeq2d 2189 . . . . . . 7  |-  ( y  =  w  ->  ( A  =  B  <->  A  =  [_ w  /  x ]_ A ) )
168, 15sbciegf 2994 . . . . . 6  |-  ( w  e.  _V  ->  ( [. w  /  y ]. A  =  B  <->  A  =  [_ w  /  x ]_ A ) )
177, 16ax-mp 5 . . . . 5  |-  ( [. w  /  y ]. A  =  B  <->  A  =  [_ w  /  x ]_ A )
18 csbeq1a 3066 . . . . . 6  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
1918eqeq1d 2186 . . . . 5  |-  ( x  =  z  ->  ( A  =  [_ w  /  x ]_ A  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
) )
2017, 19bitrid 192 . . . 4  |-  ( x  =  z  ->  ( [. w  /  y ]. A  =  B  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A ) )
216, 20sbciegf 2994 . . 3  |-  ( z  e.  _V  ->  ( [. z  /  x ]. [. w  /  y ]. A  =  B  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A ) )
223, 21ax-mp 5 . 2  |-  ( [. z  /  x ]. [. w  /  y ]. A  =  B  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
232, 22bitri 184 1  |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    e. wcel 2148   _Vcvv 2737   [.wsbc 2962   [_csb 3057   class class class wbr 4003   {copab 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065
This theorem is referenced by:  eqer  6566
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