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Theorem pwtrVD 28914
Description: Virtual deduction proof of pwtrOLD 28915. (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 4131 . . 3  |-  ( Tr 
~P A  <->  A. z A. y ( ( z  e.  y  /\  y  e.  ~P A )  -> 
z  e.  ~P A
) )
2 idn1 28641 . . . . . . 7  |-  (. Tr  A 
->.  Tr  A ).
3 idn2 28690 . . . . . . . . . 10  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  ( z  e.  y  /\  y  e.  ~P A ) ).
4 simpr 447 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  ~P A
)  ->  y  e.  ~P A )
53, 4e2 28708 . . . . . . . . 9  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  y  e.  ~P A ).
6 elpwi 3646 . . . . . . . . 9  |-  ( y  e.  ~P A  -> 
y  C_  A )
75, 6e2 28708 . . . . . . . 8  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  y  C_  A ).
8 simpl 443 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  ~P A
)  ->  z  e.  y )
93, 8e2 28708 . . . . . . . 8  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  e.  y ).
10 ssel 3187 . . . . . . . 8  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
117, 9, 10e22 28748 . . . . . . 7  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  e.  A ).
12 trss 4138 . . . . . . 7  |-  ( Tr  A  ->  ( z  e.  A  ->  z  C_  A ) )
132, 11, 12e12 28813 . . . . . 6  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  C_  A ).
14 vex 2804 . . . . . . 7  |-  z  e. 
_V
1514elpw 3644 . . . . . 6  |-  ( z  e.  ~P A  <->  z  C_  A )
1613, 15e2bir 28710 . . . . 5  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  e.  ~P A ).
1716in2 28682 . . . 4  |-  (. Tr  A 
->.  ( ( z  e.  y  /\  y  e. 
~P A )  -> 
z  e.  ~P A
) ).
1817gen12 28695 . . 3  |-  (. Tr  A 
->.  A. z A. y
( ( z  e.  y  /\  y  e. 
~P A )  -> 
z  e.  ~P A
) ).
19 bi2 189 . . 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 28768 . 2  |-  (. Tr  A 
->.  Tr  ~P A ).
2120in1 28638 1  |-  ( Tr  A  ->  Tr  ~P A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    e. wcel 1696    C_ wss 3165   ~Pcpw 3638   Tr wtr 4129
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-ral 2561  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844  df-tr 4130  df-vd1 28637  df-vd2 28646
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