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Theorem sstrALT2VD 33362
Description: Virtual deduction proof of sstrALT2 33363. (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 3478 . . 3  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
2 idn1 33079 . . . . . . 7  |-  (. ( A  C_  B  /\  B  C_  C )  ->.  ( A  C_  B  /\  B  C_  C ) ).
3 simpr 461 . . . . . . 7  |-  ( ( A  C_  B  /\  B  C_  C )  ->  B  C_  C )
42, 3e1a 33141 . . . . . 6  |-  (. ( A  C_  B  /\  B  C_  C )  ->.  B  C_  C ).
5 simpl 457 . . . . . . . 8  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  B )
62, 5e1a 33141 . . . . . . 7  |-  (. ( A  C_  B  /\  B  C_  C )  ->.  A  C_  B ).
7 idn2 33127 . . . . . . 7  |-  (. ( A  C_  B  /\  B  C_  C ) ,. x  e.  A  ->.  x  e.  A ).
8 ssel2 3484 . . . . . . 7  |-  ( ( A  C_  B  /\  x  e.  A )  ->  x  e.  B )
96, 7, 8e12an 33250 . . . . . 6  |-  (. ( A  C_  B  /\  B  C_  C ) ,. x  e.  A  ->.  x  e.  B ).
10 ssel2 3484 . . . . . 6  |-  ( ( B  C_  C  /\  x  e.  B )  ->  x  e.  C )
114, 9, 10e12an 33250 . . . . 5  |-  (. ( A  C_  B  /\  B  C_  C ) ,. x  e.  A  ->.  x  e.  C ).
1211in2 33119 . . . 4  |-  (. ( A  C_  B  /\  B  C_  C )  ->.  ( x  e.  A  ->  x  e.  C ) ).
1312gen11 33130 . . 3  |-  (. ( A  C_  B  /\  B  C_  C )  ->.  A. x
( x  e.  A  ->  x  e.  C ) ).
14 bi2 198 . . 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 33205 . 2  |-  (. ( A  C_  B  /\  B  C_  C )  ->.  A  C_  C ).
1615in1 33076 1  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1381    e. wcel 1804    C_ wss 3461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-in 3468  df-ss 3475  df-vd1 33075  df-vd2 33083
This theorem is referenced by: (None)
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