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Theorem pwtrVD 28278
Description: Virtual deduction proof of pwtr 4357. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
pwtrVD  |-  ( Tr  A  ->  Tr  ~P A
)

Proof of Theorem pwtrVD
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 4245 . . 3  |-  ( Tr 
~P A  <->  A. z A. y ( ( z  e.  y  /\  y  e.  ~P A )  -> 
z  e.  ~P A
) )
2 idn1 28006 . . . . . . 7  |-  (. Tr  A 
->.  Tr  A ).
3 idn2 28055 . . . . . . . . . 10  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  ( z  e.  y  /\  y  e.  ~P A ) ).
4 simpr 448 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  ~P A
)  ->  y  e.  ~P A )
53, 4e2 28073 . . . . . . . . 9  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  y  e.  ~P A ).
6 elpwi 3750 . . . . . . . . 9  |-  ( y  e.  ~P A  -> 
y  C_  A )
75, 6e2 28073 . . . . . . . 8  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  y  C_  A ).
8 simpl 444 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  ~P A
)  ->  z  e.  y )
93, 8e2 28073 . . . . . . . 8  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  e.  y ).
10 ssel 3285 . . . . . . . 8  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
117, 9, 10e22 28113 . . . . . . 7  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  e.  A ).
12 trss 4252 . . . . . . 7  |-  ( Tr  A  ->  ( z  e.  A  ->  z  C_  A ) )
132, 11, 12e12 28177 . . . . . 6  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  C_  A ).
14 vex 2902 . . . . . . 7  |-  z  e. 
_V
1514elpw 3748 . . . . . 6  |-  ( z  e.  ~P A  <->  z  C_  A )
1613, 15e2bir 28075 . . . . 5  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  e.  ~P A ).
1716in2 28047 . . . 4  |-  (. Tr  A 
->.  ( ( z  e.  y  /\  y  e. 
~P A )  -> 
z  e.  ~P A
) ).
1817gen12 28060 . . 3  |-  (. Tr  A 
->.  A. z A. y
( ( z  e.  y  /\  y  e. 
~P A )  -> 
z  e.  ~P A
) ).
19 bi2 190 . . 3  |-  ( ( Tr  ~P A  <->  A. z A. y ( ( z  e.  y  /\  y  e.  ~P A )  -> 
z  e.  ~P A
) )  ->  ( A. z A. y ( ( z  e.  y  /\  y  e.  ~P A )  ->  z  e.  ~P A )  ->  Tr  ~P A ) )
201, 18, 19e01 28133 . 2  |-  (. Tr  A 
->.  Tr  ~P A ).
2120in1 28003 1  |-  ( Tr  A  ->  Tr  ~P A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    e. wcel 1717    C_ wss 3263   ~Pcpw 3742   Tr wtr 4243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-v 2901  df-in 3270  df-ss 3277  df-pw 3744  df-uni 3958  df-tr 4244  df-vd1 28002  df-vd2 28011
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