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Theorem csbeq2gVD 29004
Description: Virtual deduction proof of csbeq2g 28636. 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 28636 is csbeq2gVD 29004 without virtual deductions and was automatically derived from csbeq2gVD 29004.
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 28665 . . . 4  |-  (. A  e.  V  ->.  A  e.  V ).
2 spsbc 3173 . . . 4  |-  ( A  e.  V  ->  ( A. x  B  =  C  ->  [. A  /  x ]. B  =  C
) )
31, 2e1_ 28728 . . 3  |-  (. A  e.  V  ->.  ( A. x  B  =  C  ->  [. A  /  x ]. B  =  C ) ).
4 sbceqg 3267 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
51, 4e1_ 28728 . . 3  |-  (. A  e.  V  ->.  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) ).
6 imbi2 315 . . . 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 216 . . 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 28789 . 2  |-  (. A  e.  V  ->.  ( A. x  B  =  C  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) ).
98in1 28662 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 177   A.wal 1549    = wceq 1652    e. wcel 1725   [.wsbc 3161   [_csb 3251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162  df-csb 3252  df-vd1 28661
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