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Theorem eqerlem 7138
Description: Lemma for eqer 7139. (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 4605 . 2  |-  ( z R w  <->  [. z  /  x ]. [. w  / 
y ]. A  =  B )
3 vex 2980 . . 3  |-  z  e. 
_V
4 nfcsb1v 3309 . . . . 5  |-  F/_ x [_ z  /  x ]_ A
5 nfcsb1v 3309 . . . . 5  |-  F/_ x [_ w  /  x ]_ A
64, 5nfeq 2591 . . . 4  |-  F/ x [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
7 vex 2980 . . . . . 6  |-  w  e. 
_V
8 nfv 1673 . . . . . . 7  |-  F/ y  A  =  [_ w  /  x ]_ A
9 vex 2980 . . . . . . . . . 10  |-  y  e. 
_V
10 nfcv 2584 . . . . . . . . . 10  |-  F/_ x B
11 eqer.1 . . . . . . . . . 10  |-  ( x  =  y  ->  A  =  B )
129, 10, 11csbief 3318 . . . . . . . . 9  |-  [_ y  /  x ]_ A  =  B
13 csbeq1 3296 . . . . . . . . 9  |-  ( y  =  w  ->  [_ y  /  x ]_ A  = 
[_ w  /  x ]_ A )
1412, 13syl5eqr 2489 . . . . . . . 8  |-  ( y  =  w  ->  B  =  [_ w  /  x ]_ A )
1514eqeq2d 2454 . . . . . . 7  |-  ( y  =  w  ->  ( A  =  B  <->  A  =  [_ w  /  x ]_ A ) )
168, 15sbciegf 3223 . . . . . 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 3302 . . . . . 6  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
1918eqeq1d 2451 . . . . 5  |-  ( x  =  z  ->  ( A  =  [_ w  /  x ]_ A  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
) )
2017, 19syl5bb 257 . . . 4  |-  ( x  =  z  ->  ( [. w  /  y ]. A  =  B  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A ) )
216, 20sbciegf 3223 . . 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 249 1  |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   _Vcvv 2977   [.wsbc 3191   [_csb 3293   class class class wbr 4297   {copab 4354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356
This theorem is referenced by:  eqer  7139
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