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Theorem csbcomg 3117
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 2809 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2809 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 sbccom 3075 . . . . . 6  |-  ( [. A  /  x ]. [. B  /  y ]. z  e.  C  <->  [. B  /  y ]. [. A  /  x ]. z  e.  C
)
43a1i 10 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. [. B  / 
y ]. z  e.  C  <->  [. B  /  y ]. [. A  /  x ]. z  e.  C )
)
5 sbcel2g 3115 . . . . . . 7  |-  ( B  e.  _V  ->  ( [. B  /  y ]. z  e.  C  <->  z  e.  [_ B  / 
y ]_ C ) )
65sbcbidv 3058 . . . . . 6  |-  ( B  e.  _V  ->  ( [. A  /  x ]. [. B  /  y ]. z  e.  C  <->  [. A  /  x ]. z  e.  [_ B  / 
y ]_ C ) )
76adantl 452 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. [. B  / 
y ]. z  e.  C  <->  [. A  /  x ]. z  e.  [_ B  / 
y ]_ C ) )
8 sbcel2g 3115 . . . . . . 7  |-  ( A  e.  _V  ->  ( [. A  /  x ]. z  e.  C  <->  z  e.  [_ A  /  x ]_ C ) )
98sbcbidv 3058 . . . . . 6  |-  ( A  e.  _V  ->  ( [. B  /  y ]. [. A  /  x ]. z  e.  C  <->  [. B  /  y ]. z  e.  [_ A  /  x ]_ C ) )
109adantr 451 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. B  / 
y ]. [. A  /  x ]. z  e.  C  <->  [. B  /  y ]. z  e.  [_ A  /  x ]_ C ) )
114, 7, 103bitr3d 274 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  [. B  /  y ]. z  e.  [_ A  /  x ]_ C ) )
12 sbcel2g 3115 . . . . 5  |-  ( A  e.  _V  ->  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  z  e.  [_ A  /  x ]_ [_ B  / 
y ]_ C ) )
1312adantr 451 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  z  e.  [_ A  /  x ]_ [_ B  / 
y ]_ C ) )
14 sbcel2g 3115 . . . . 5  |-  ( B  e.  _V  ->  ( [. B  /  y ]. z  e.  [_ A  /  x ]_ C  <->  z  e.  [_ B  /  y ]_ [_ A  /  x ]_ C ) )
1514adantl 452 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. B  / 
y ]. z  e.  [_ A  /  x ]_ C  <->  z  e.  [_ B  / 
y ]_ [_ A  /  x ]_ C ) )
1611, 13, 153bitr3d 274 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( z  e.  [_ A  /  x ]_ [_ B  /  y ]_ C  <->  z  e.  [_ B  / 
y ]_ [_ A  /  x ]_ C ) )
1716eqrdv 2294 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  [_ B  /  y ]_ [_ A  /  x ]_ C )
181, 2, 17syl2an 463 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    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   [.wsbc 3004   [_csb 3094
This theorem is referenced by:  ovmpt2s  5987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005  df-csb 3095
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