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

Theorem eqerlem 6428
Description: Lemma for eqer 6429. (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 4153 . 2  |-  ( z R w  <->  [. z  /  x ]. [. w  / 
y ]. A  =  B )
3 vex 2663 . . 3  |-  z  e. 
_V
4 nfcsb1v 3005 . . . . 5  |-  F/_ x [_ z  /  x ]_ A
5 nfcsb1v 3005 . . . . 5  |-  F/_ x [_ w  /  x ]_ A
64, 5nfeq 2266 . . . 4  |-  F/ x [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
7 vex 2663 . . . . . 6  |-  w  e. 
_V
8 nfv 1493 . . . . . . 7  |-  F/ y  A  =  [_ w  /  x ]_ A
9 vex 2663 . . . . . . . . . 10  |-  y  e. 
_V
10 nfcv 2258 . . . . . . . . . 10  |-  F/_ x B
11 eqer.1 . . . . . . . . . 10  |-  ( x  =  y  ->  A  =  B )
129, 10, 11csbief 3014 . . . . . . . . 9  |-  [_ y  /  x ]_ A  =  B
13 csbeq1 2978 . . . . . . . . 9  |-  ( y  =  w  ->  [_ y  /  x ]_ A  = 
[_ w  /  x ]_ A )
1412, 13syl5eqr 2164 . . . . . . . 8  |-  ( y  =  w  ->  B  =  [_ w  /  x ]_ A )
1514eqeq2d 2129 . . . . . . 7  |-  ( y  =  w  ->  ( A  =  B  <->  A  =  [_ w  /  x ]_ A ) )
168, 15sbciegf 2912 . . . . . 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 2983 . . . . . 6  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
1918eqeq1d 2126 . . . . 5  |-  ( x  =  z  ->  ( A  =  [_ w  /  x ]_ A  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
) )
2017, 19syl5bb 191 . . . 4  |-  ( x  =  z  ->  ( [. w  /  y ]. A  =  B  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A ) )
216, 20sbciegf 2912 . . 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 183 1  |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316    e. wcel 1465   _Vcvv 2660   [.wsbc 2882   [_csb 2975   class class class wbr 3899   {copab 3958
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960
This theorem is referenced by:  eqer  6429
  Copyright terms: Public domain W3C validator