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Theorem csbeq2gVD 28668
Description: Virtual deduction proof of csbeq2g 28315. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbeq2g 28315 is csbeq2gVD 28668 without virtual deductions and was automatically derived from csbeq2gVD 28668.
1::  |-  (. A  e.  V  ->.  A  e.  V ).
2:1:  |-  (. A  e.  V  ->.  ( A. x B  =  C  ->  [. A  /  x ].  B  =  C ) ).
3:1:  |-  (. A  e.  V  ->.  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) ).
4:2,3:  |-  (. A  e.  V  ->.  ( A. x B  =  C  ->  [_ A  /  x  ]_ B  =  [_ A  /  x ]_ C ) ).
qed:4:  |-  ( A  e.  V  ->  ( A. x B  =  C  ->  [_ A  /  x ]_  B  =  [_ A  /  x ]_ C ) )
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbeq2gVD  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )

Proof of Theorem csbeq2gVD
StepHypRef Expression
1 idn1 28342 . . . 4  |-  (. A  e.  V  ->.  A  e.  V ).
2 spsbc 3003 . . . 4  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [. A  /  x ]. B  =  C
) )
31, 2e1_ 28399 . . 3  |-  (. A  e.  V  ->.  ( A. x  B  =  C  ->  [. A  /  x ]. B  =  C ) ).
4 sbceqg 3097 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
51, 4e1_ 28399 . . 3  |-  (. A  e.  V  ->.  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) ).
6 imbi2 314 . . . 4  |-  ( (
[. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )  -> 
( ( A. x  B  =  C  ->  [. A  /  x ]. B  =  C )  <->  ( A. x  B  =  C  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C
) ) )
76biimpcd 215 . . 3  |-  ( ( A. x  B  =  C  ->  [. A  /  x ]. B  =  C )  ->  ( ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )  -> 
( A. x  B  =  C  ->  [_ A  /  x ]_ B  = 
[_ A  /  x ]_ C ) ) )
83, 5, 7e11 28460 . 2  |-  (. A  e.  V  ->.  ( A. x  B  =  C  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) ).
98in1 28339 1  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684   [.wsbc 2991   [_csb 3081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082  df-vd1 28338
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