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Theorem sstrALT2VD 28946
Description: Virtual deduction proof of sstrALT2 28947. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sstrALT2VD  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )

Proof of Theorem sstrALT2VD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dfss2 3337 . . 3  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
2 idn1 28665 . . . . . . 7  |-  (. ( A  C_  B  /\  B  C_  C )  ->.  ( A  C_  B  /\  B  C_  C ) ).
3 simpr 448 . . . . . . 7  |-  ( ( A  C_  B  /\  B  C_  C )  ->  B  C_  C )
42, 3e1_ 28728 . . . . . 6  |-  (. ( A  C_  B  /\  B  C_  C )  ->.  B  C_  C ).
5 simpl 444 . . . . . . . 8  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  B )
62, 5e1_ 28728 . . . . . . 7  |-  (. ( A  C_  B  /\  B  C_  C )  ->.  A  C_  B ).
7 idn2 28714 . . . . . . 7  |-  (. ( A  C_  B  /\  B  C_  C ) ,. x  e.  A  ->.  x  e.  A ).
8 ssel2 3343 . . . . . . 7  |-  ( ( A  C_  B  /\  x  e.  A )  ->  x  e.  B )
96, 7, 8e12an 28837 . . . . . 6  |-  (. ( A  C_  B  /\  B  C_  C ) ,. x  e.  A  ->.  x  e.  B ).
10 ssel2 3343 . . . . . 6  |-  ( ( B  C_  C  /\  x  e.  B )  ->  x  e.  C )
114, 9, 10e12an 28837 . . . . 5  |-  (. ( A  C_  B  /\  B  C_  C ) ,. x  e.  A  ->.  x  e.  C ).
1211in2 28706 . . . 4  |-  (. ( A  C_  B  /\  B  C_  C )  ->.  ( x  e.  A  ->  x  e.  C ) ).
1312gen11 28717 . . 3  |-  (. ( A  C_  B  /\  B  C_  C )  ->.  A. x
( x  e.  A  ->  x  e.  C ) ).
14 bi2 190 . . 3  |-  ( ( A  C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )  ->  ( A. x ( x  e.  A  ->  x  e.  C )  ->  A  C_  C ) )
151, 13, 14e01 28792 . 2  |-  (. ( A  C_  B  /\  B  C_  C )  ->.  A  C_  C ).
1615in1 28662 1  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    e. wcel 1725    C_ wss 3320
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-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-in 3327  df-ss 3334  df-vd1 28661  df-vd2 28670
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