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Theorem csbcomg 3276
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 2966 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2966 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 sbccom 3234 . . . . . 6  |-  ( [. A  /  x ]. [. B  /  y ]. z  e.  C  <->  [. B  /  y ]. [. A  /  x ]. z  e.  C
)
43a1i 11 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. [. B  / 
y ]. z  e.  C  <->  [. B  /  y ]. [. A  /  x ]. z  e.  C )
)
5 sbcel2g 3274 . . . . . . 7  |-  ( B  e.  _V  ->  ( [. B  /  y ]. z  e.  C  <->  z  e.  [_ B  / 
y ]_ C ) )
65sbcbidv 3217 . . . . . 6  |-  ( B  e.  _V  ->  ( [. A  /  x ]. [. B  /  y ]. z  e.  C  <->  [. A  /  x ]. z  e.  [_ B  / 
y ]_ C ) )
76adantl 454 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. [. B  / 
y ]. z  e.  C  <->  [. A  /  x ]. z  e.  [_ B  / 
y ]_ C ) )
8 sbcel2g 3274 . . . . . . 7  |-  ( A  e.  _V  ->  ( [. A  /  x ]. z  e.  C  <->  z  e.  [_ A  /  x ]_ C ) )
98sbcbidv 3217 . . . . . 6  |-  ( A  e.  _V  ->  ( [. B  /  y ]. [. A  /  x ]. z  e.  C  <->  [. B  /  y ]. z  e.  [_ A  /  x ]_ C ) )
109adantr 453 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. B  / 
y ]. [. A  /  x ]. z  e.  C  <->  [. B  /  y ]. z  e.  [_ A  /  x ]_ C ) )
114, 7, 103bitr3d 276 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  [. B  /  y ]. z  e.  [_ A  /  x ]_ C ) )
12 sbcel2g 3274 . . . . 5  |-  ( A  e.  _V  ->  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  z  e.  [_ A  /  x ]_ [_ B  / 
y ]_ C ) )
1312adantr 453 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  z  e.  [_ A  /  x ]_ [_ B  / 
y ]_ C ) )
14 sbcel2g 3274 . . . . 5  |-  ( B  e.  _V  ->  ( [. B  /  y ]. z  e.  [_ A  /  x ]_ C  <->  z  e.  [_ B  /  y ]_ [_ A  /  x ]_ C ) )
1514adantl 454 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. B  / 
y ]. z  e.  [_ A  /  x ]_ C  <->  z  e.  [_ B  / 
y ]_ [_ A  /  x ]_ C ) )
1611, 13, 153bitr3d 276 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( z  e.  [_ A  /  x ]_ [_ B  /  y ]_ C  <->  z  e.  [_ B  / 
y ]_ [_ A  /  x ]_ C ) )
1716eqrdv 2436 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  [_ B  /  y ]_ [_ A  /  x ]_ C )
181, 2, 17syl2an 465 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   [.wsbc 3163   [_csb 3253
This theorem is referenced by:  ovmpt2s  6199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164  df-csb 3254
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