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Theorem csbeq2gVD 37289
Description: Virtual deduction proof of csbeq2gOLD 36916. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbeq2gOLD 36916 is csbeq2gVD 37289 without virtual deductions and was automatically derived from csbeq2gVD 37289.
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 36944 . . . 4  |-  (. A  e.  V  ->.  A  e.  V ).
2 spsbc 3280 . . . 4  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [. A  /  x ]. B  =  C
) )
31, 2e1a 37006 . . 3  |-  (. A  e.  V  ->.  ( A. x  B  =  C  ->  [. A  /  x ]. B  =  C ) ).
4 sbceqg 3773 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
51, 4e1a 37006 . . 3  |-  (. A  e.  V  ->.  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) ).
6 imbi2 326 . . . 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 228 . . 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 37067 . 2  |-  (. A  e.  V  ->.  ( A. x  B  =  C  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) ).
98in1 36941 1  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1442    = wceq 1444    e. wcel 1887   [.wsbc 3267   [_csb 3363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-sbc 3268  df-csb 3364  df-vd1 36940
This theorem is referenced by: (None)
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