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Theorem csbcomg 2954
Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
Assertion
Ref Expression
csbcomg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  [_ B  /  y ]_ [_ A  /  x ]_ C )
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    A( x)    B( y)    C( x, y)    V( x, y)    W( x, y)

Proof of Theorem csbcomg
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elex 2630 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2630 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 sbccom 2914 . . . . . 6  |-  ( [. A  /  x ]. [. B  /  y ]. z  e.  C  <->  [. B  /  y ]. [. A  /  x ]. z  e.  C
)
43a1i 9 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. [. B  / 
y ]. z  e.  C  <->  [. B  /  y ]. [. A  /  x ]. z  e.  C )
)
5 sbcel2g 2952 . . . . . . 7  |-  ( B  e.  _V  ->  ( [. B  /  y ]. z  e.  C  <->  z  e.  [_ B  / 
y ]_ C ) )
65sbcbidv 2897 . . . . . 6  |-  ( B  e.  _V  ->  ( [. A  /  x ]. [. B  /  y ]. z  e.  C  <->  [. A  /  x ]. z  e.  [_ B  / 
y ]_ C ) )
76adantl 271 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. [. B  / 
y ]. z  e.  C  <->  [. A  /  x ]. z  e.  [_ B  / 
y ]_ C ) )
8 sbcel2g 2952 . . . . . . 7  |-  ( A  e.  _V  ->  ( [. A  /  x ]. z  e.  C  <->  z  e.  [_ A  /  x ]_ C ) )
98sbcbidv 2897 . . . . . 6  |-  ( A  e.  _V  ->  ( [. B  /  y ]. [. A  /  x ]. z  e.  C  <->  [. B  /  y ]. z  e.  [_ A  /  x ]_ C ) )
109adantr 270 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. B  / 
y ]. [. A  /  x ]. z  e.  C  <->  [. B  /  y ]. z  e.  [_ A  /  x ]_ C ) )
114, 7, 103bitr3d 216 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  [. B  /  y ]. z  e.  [_ A  /  x ]_ C ) )
12 sbcel2g 2952 . . . . 5  |-  ( A  e.  _V  ->  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  z  e.  [_ A  /  x ]_ [_ B  / 
y ]_ C ) )
1312adantr 270 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  z  e.  [_ A  /  x ]_ [_ B  / 
y ]_ C ) )
14 sbcel2g 2952 . . . . 5  |-  ( B  e.  _V  ->  ( [. B  /  y ]. z  e.  [_ A  /  x ]_ C  <->  z  e.  [_ B  /  y ]_ [_ A  /  x ]_ C ) )
1514adantl 271 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. B  / 
y ]. z  e.  [_ A  /  x ]_ C  <->  z  e.  [_ B  / 
y ]_ [_ A  /  x ]_ C ) )
1611, 13, 153bitr3d 216 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( z  e.  [_ A  /  x ]_ [_ B  /  y ]_ C  <->  z  e.  [_ B  / 
y ]_ [_ A  /  x ]_ C ) )
1716eqrdv 2086 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  [_ B  /  y ]_ [_ A  /  x ]_ C )
181, 2, 17syl2an 283 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  [_ B  /  y ]_ [_ A  /  x ]_ C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   _Vcvv 2619   [.wsbc 2840   [_csb 2933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-csb 2934
This theorem is referenced by:  ovmpt2s  5768
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