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Theorem eqerlem 7345
Description: Lemma for eqer 7346. (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 4748 . 2  |-  ( z R w  <->  [. z  /  x ]. [. w  / 
y ]. A  =  B )
3 vex 3098 . . 3  |-  z  e. 
_V
4 nfcsb1v 3436 . . . . 5  |-  F/_ x [_ z  /  x ]_ A
5 nfcsb1v 3436 . . . . 5  |-  F/_ x [_ w  /  x ]_ A
64, 5nfeq 2616 . . . 4  |-  F/ x [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
7 vex 3098 . . . . . 6  |-  w  e. 
_V
8 nfv 1694 . . . . . . 7  |-  F/ y  A  =  [_ w  /  x ]_ A
9 vex 3098 . . . . . . . . . 10  |-  y  e. 
_V
10 eqer.1 . . . . . . . . . 10  |-  ( x  =  y  ->  A  =  B )
119, 10csbie 3446 . . . . . . . . 9  |-  [_ y  /  x ]_ A  =  B
12 csbeq1 3423 . . . . . . . . 9  |-  ( y  =  w  ->  [_ y  /  x ]_ A  = 
[_ w  /  x ]_ A )
1311, 12syl5eqr 2498 . . . . . . . 8  |-  ( y  =  w  ->  B  =  [_ w  /  x ]_ A )
1413eqeq2d 2457 . . . . . . 7  |-  ( y  =  w  ->  ( A  =  B  <->  A  =  [_ w  /  x ]_ A ) )
158, 14sbciegf 3345 . . . . . 6  |-  ( w  e.  _V  ->  ( [. w  /  y ]. A  =  B  <->  A  =  [_ w  /  x ]_ A ) )
167, 15ax-mp 5 . . . . 5  |-  ( [. w  /  y ]. A  =  B  <->  A  =  [_ w  /  x ]_ A )
17 csbeq1a 3429 . . . . . 6  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
1817eqeq1d 2445 . . . . 5  |-  ( x  =  z  ->  ( A  =  [_ w  /  x ]_ A  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
) )
1916, 18syl5bb 257 . . . 4  |-  ( x  =  z  ->  ( [. w  /  y ]. A  =  B  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A ) )
206, 19sbciegf 3345 . . 3  |-  ( z  e.  _V  ->  ( [. z  /  x ]. [. w  /  y ]. A  =  B  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A ) )
213, 20ax-mp 5 . 2  |-  ( [. z  /  x ]. [. w  /  y ]. A  =  B  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
222, 21bitri 249 1  |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1383    e. wcel 1804   _Vcvv 3095   [.wsbc 3313   [_csb 3420   class class class wbr 4437   {copab 4494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496
This theorem is referenced by:  eqer  7346
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