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Theorem eqsbc3rVD 28932
Description: Virtual deduction proof of eqsbc3r 3061. (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 28641 . . . . . . 7  |-  (. A  e.  B  ->.  A  e.  B ).
2 eqsbc3 3043 . . . . . . 7  |-  ( A  e.  B  ->  ( [. A  /  x ]. x  =  C  <->  A  =  C ) )
31, 2e1_ 28704 . . . . . 6  |-  (. A  e.  B  ->.  ( [. A  /  x ]. x  =  C  <->  A  =  C
) ).
4 eqcom 2298 . . . . . . . . 9  |-  ( C  =  x  <->  x  =  C )
54sbcbiiOLD 3060 . . . . . . . 8  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  <->  [. A  /  x ]. x  =  C )
)
61, 5e1_ 28704 . . . . . . 7  |-  (. A  e.  B  ->.  ( [. A  /  x ]. C  =  x  <->  [. A  /  x ]. x  =  C
) ).
7 idn2 28690 . . . . . . 7  |-  (. A  e.  B ,. [. A  /  x ]. C  =  x  ->.  [. A  /  x ]. C  =  x ).
8 bi1 178 . . . . . . 7  |-  ( (
[. A  /  x ]. C  =  x  <->  [. A  /  x ]. x  =  C )  ->  ( [. A  /  x ]. C  =  x  ->  [. A  /  x ]. x  =  C
) )
96, 7, 8e12 28813 . . . . . 6  |-  (. A  e.  B ,. [. A  /  x ]. C  =  x  ->.  [. A  /  x ]. x  =  C ).
10 bi1 178 . . . . . 6  |-  ( (
[. A  /  x ]. x  =  C  <->  A  =  C )  -> 
( [. A  /  x ]. x  =  C  ->  A  =  C ) )
113, 9, 10e12 28813 . . . . 5  |-  (. A  e.  B ,. [. A  /  x ]. C  =  x  ->.  A  =  C ).
12 eqcom 2298 . . . . 5  |-  ( A  =  C  <->  C  =  A )
1311, 12e2bi 28709 . . . 4  |-  (. A  e.  B ,. [. A  /  x ]. C  =  x  ->.  C  =  A ).
1413in2 28682 . . 3  |-  (. A  e.  B  ->.  ( [. A  /  x ]. C  =  x  ->  C  =  A ) ).
15 idn2 28690 . . . . . . 7  |-  (. A  e.  B ,. C  =  A  ->.  C  =  A ).
1615, 12e2bir 28710 . . . . . 6  |-  (. A  e.  B ,. C  =  A  ->.  A  =  C ).
17 bi2 189 . . . . . 6  |-  ( (
[. A  /  x ]. x  =  C  <->  A  =  C )  -> 
( A  =  C  ->  [. A  /  x ]. x  =  C
) )
183, 16, 17e12 28813 . . . . 5  |-  (. A  e.  B ,. C  =  A  ->.  [. A  /  x ]. x  =  C ).
19 bi2 189 . . . . 5  |-  ( (
[. A  /  x ]. C  =  x  <->  [. A  /  x ]. x  =  C )  ->  ( [. A  /  x ]. x  =  C  ->  [. A  /  x ]. C  =  x
) )
206, 18, 19e12 28813 . . . 4  |-  (. A  e.  B ,. C  =  A  ->.  [. A  /  x ]. C  =  x ).
2120in2 28682 . . 3  |-  (. A  e.  B  ->.  ( C  =  A  ->  [. A  /  x ]. C  =  x ) ).
22 bi3 179 . . 3  |-  ( (
[. A  /  x ]. C  =  x  ->  C  =  A )  ->  ( ( C  =  A  ->  [. A  /  x ]. C  =  x )  ->  ( [. A  /  x ]. C  =  x  <->  C  =  A ) ) )
2314, 21, 22e11 28765 . 2  |-  (. A  e.  B  ->.  ( [. A  /  x ]. C  =  x  <->  C  =  A
) ).
2423in1 28638 1  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  <->  C  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   [.wsbc 3004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005  df-vd1 28637  df-vd2 28646
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