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Theorem eqsbc3rVD 37230
Description: Virtual deduction proof of eqsbc3r 3323. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
eqsbc3rVD  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  <->  C  =  A ) )
Distinct variable groups:    x, C    x, A
Allowed substitution hint:    B( x)

Proof of Theorem eqsbc3rVD
StepHypRef Expression
1 idn1 36938 . . . . . . 7  |-  (. A  e.  B  ->.  A  e.  B ).
2 eqsbc3 3306 . . . . . . 7  |-  ( A  e.  B  ->  ( [. A  /  x ]. x  =  C  <->  A  =  C ) )
31, 2e1a 37000 . . . . . 6  |-  (. A  e.  B  ->.  ( [. A  /  x ]. x  =  C  <->  A  =  C
) ).
4 eqcom 2457 . . . . . . . . 9  |-  ( C  =  x  <->  x  =  C )
54sbcbiiOLD 36886 . . . . . . . 8  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  <->  [. A  /  x ]. x  =  C )
)
61, 5e1a 37000 . . . . . . 7  |-  (. A  e.  B  ->.  ( [. A  /  x ]. C  =  x  <->  [. A  /  x ]. x  =  C
) ).
7 idn2 36986 . . . . . . 7  |-  (. A  e.  B ,. [. A  /  x ]. C  =  x  ->.  [. A  /  x ]. C  =  x ).
8 biimp 197 . . . . . . 7  |-  ( (
[. A  /  x ]. C  =  x  <->  [. A  /  x ]. x  =  C )  ->  ( [. A  /  x ]. C  =  x  ->  [. A  /  x ]. x  =  C
) )
96, 7, 8e12 37105 . . . . . 6  |-  (. A  e.  B ,. [. A  /  x ]. C  =  x  ->.  [. A  /  x ]. x  =  C ).
10 biimp 197 . . . . . 6  |-  ( (
[. A  /  x ]. x  =  C  <->  A  =  C )  -> 
( [. A  /  x ]. x  =  C  ->  A  =  C ) )
113, 9, 10e12 37105 . . . . 5  |-  (. A  e.  B ,. [. A  /  x ]. C  =  x  ->.  A  =  C ).
12 eqcom 2457 . . . . 5  |-  ( A  =  C  <->  C  =  A )
1311, 12e2bi 37005 . . . 4  |-  (. A  e.  B ,. [. A  /  x ]. C  =  x  ->.  C  =  A ).
1413in2 36978 . . 3  |-  (. A  e.  B  ->.  ( [. A  /  x ]. C  =  x  ->  C  =  A ) ).
15 idn2 36986 . . . . . . 7  |-  (. A  e.  B ,. C  =  A  ->.  C  =  A ).
1615, 12e2bir 37006 . . . . . 6  |-  (. A  e.  B ,. C  =  A  ->.  A  =  C ).
17 biimpr 202 . . . . . 6  |-  ( (
[. A  /  x ]. x  =  C  <->  A  =  C )  -> 
( A  =  C  ->  [. A  /  x ]. x  =  C
) )
183, 16, 17e12 37105 . . . . 5  |-  (. A  e.  B ,. C  =  A  ->.  [. A  /  x ]. x  =  C ).
19 biimpr 202 . . . . 5  |-  ( (
[. A  /  x ]. C  =  x  <->  [. A  /  x ]. x  =  C )  ->  ( [. A  /  x ]. x  =  C  ->  [. A  /  x ]. C  =  x
) )
206, 18, 19e12 37105 . . . 4  |-  (. A  e.  B ,. C  =  A  ->.  [. A  /  x ]. C  =  x ).
2120in2 36978 . . 3  |-  (. A  e.  B  ->.  ( C  =  A  ->  [. A  /  x ]. C  =  x ) ).
22 impbi 190 . . 3  |-  ( (
[. A  /  x ]. C  =  x  ->  C  =  A )  ->  ( ( C  =  A  ->  [. A  /  x ]. C  =  x )  ->  ( [. A  /  x ]. C  =  x  <->  C  =  A ) ) )
2314, 21, 22e11 37061 . 2  |-  (. A  e.  B  ->.  ( [. A  /  x ]. C  =  x  <->  C  =  A
) ).
2423in1 36935 1  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  <->  C  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    = wceq 1443    e. wcel 1886   [.wsbc 3266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3046  df-sbc 3267  df-vd1 36934  df-vd2 36942
This theorem is referenced by: (None)
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