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

Theorem eqerlem 6168
Description: Lemma for eqer 6169. (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 4026 . 2  |-  ( z R w  <->  [. z  /  x ]. [. w  / 
y ]. A  =  B )
3 vex 2577 . . 3  |-  z  e. 
_V
4 nfcsb1v 2910 . . . . 5  |-  F/_ x [_ z  /  x ]_ A
5 nfcsb1v 2910 . . . . 5  |-  F/_ x [_ w  /  x ]_ A
64, 5nfeq 2201 . . . 4  |-  F/ x [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
7 vex 2577 . . . . . 6  |-  w  e. 
_V
8 nfv 1437 . . . . . . 7  |-  F/ y  A  =  [_ w  /  x ]_ A
9 vex 2577 . . . . . . . . . 10  |-  y  e. 
_V
10 nfcv 2194 . . . . . . . . . 10  |-  F/_ x B
11 eqer.1 . . . . . . . . . 10  |-  ( x  =  y  ->  A  =  B )
129, 10, 11csbief 2919 . . . . . . . . 9  |-  [_ y  /  x ]_ A  =  B
13 csbeq1 2883 . . . . . . . . 9  |-  ( y  =  w  ->  [_ y  /  x ]_ A  = 
[_ w  /  x ]_ A )
1412, 13syl5eqr 2102 . . . . . . . 8  |-  ( y  =  w  ->  B  =  [_ w  /  x ]_ A )
1514eqeq2d 2067 . . . . . . 7  |-  ( y  =  w  ->  ( A  =  B  <->  A  =  [_ w  /  x ]_ A ) )
168, 15sbciegf 2817 . . . . . 6  |-  ( w  e.  _V  ->  ( [. w  /  y ]. A  =  B  <->  A  =  [_ w  /  x ]_ A ) )
177, 16ax-mp 7 . . . . 5  |-  ( [. w  /  y ]. A  =  B  <->  A  =  [_ w  /  x ]_ A )
18 csbeq1a 2888 . . . . . 6  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
1918eqeq1d 2064 . . . . 5  |-  ( x  =  z  ->  ( A  =  [_ w  /  x ]_ A  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
) )
2017, 19syl5bb 185 . . . 4  |-  ( x  =  z  ->  ( [. w  /  y ]. A  =  B  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A ) )
216, 20sbciegf 2817 . . 3  |-  ( z  e.  _V  ->  ( [. z  /  x ]. [. w  /  y ]. A  =  B  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A ) )
223, 21ax-mp 7 . 2  |-  ( [. z  /  x ]. [. w  /  y ]. A  =  B  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
232, 22bitri 177 1  |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259    e. wcel 1409   _Vcvv 2574   [.wsbc 2787   [_csb 2880   class class class wbr 3792   {copab 3845
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-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847
This theorem is referenced by:  eqer  6169
  Copyright terms: Public domain W3C validator