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Theorem csbcomg 2867
Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
Assertion
Ref Expression
csbcomg  V  W  [_  ]_
[_  ]_ C  [_  ]_ [_  ]_ C
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()    C(,)    V(,)    W(,)

Proof of Theorem csbcomg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2560 . 2  V  _V
2 elex 2560 . 2  W  _V
3 sbccom 2827 . . . . . 6  [.  ]. [.  ].  C  [.  ]. [.  ].  C
43a1i 9 . . . . 5  _V  _V  [.  ]. [.  ].  C  [.  ]. [.  ].  C
5 sbcel2g 2865 . . . . . . 7  _V  [.  ].  C  [_  ]_ C
65sbcbidv 2811 . . . . . 6  _V  [.  ].
[.  ].  C  [.  ].  [_  ]_ C
76adantl 262 . . . . 5  _V  _V  [.  ]. [.  ].  C  [.  ].  [_  ]_ C
8 sbcel2g 2865 . . . . . . 7  _V  [.  ].  C 
[_  ]_ C
98sbcbidv 2811 . . . . . 6  _V  [.  ]. [.  ].  C  [.  ]. 
[_  ]_ C
109adantr 261 . . . . 5  _V  _V  [.  ]. [.  ].  C  [.  ].  [_  ]_ C
114, 7, 103bitr3d 207 . . . 4  _V  _V  [.  ].  [_  ]_ C  [.  ].  [_  ]_ C
12 sbcel2g 2865 . . . . 5  _V  [.  ].  [_  ]_ C 
[_  ]_
[_  ]_ C
1312adantr 261 . . . 4  _V  _V  [.  ].  [_  ]_ C  [_  ]_ [_  ]_ C
14 sbcel2g 2865 . . . . 5  _V  [.  ].  [_  ]_ C 
[_  ]_ [_  ]_ C
1514adantl 262 . . . 4  _V  _V  [.  ].  [_  ]_ C  [_  ]_ [_  ]_ C
1611, 13, 153bitr3d 207 . . 3  _V  _V  [_  ]_ [_  ]_ C 
[_  ]_ [_  ]_ C
1716eqrdv 2035 . 2  _V  _V  [_  ]_
[_  ]_ C  [_  ]_ [_  ]_ C
181, 2, 17syl2an 273 1  V  W  [_  ]_
[_  ]_ C  [_  ]_ [_  ]_ C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390   _Vcvv 2551   [.wsbc 2758   [_csb 2846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847
This theorem is referenced by:  ovmpt2s  5566
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