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Theorem pwtrrVD 28280
Description: Virtual deduction proof of pwtr 4358. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
pwtrrVD.1  |-  A  e. 
_V
Assertion
Ref Expression
pwtrrVD  |-  ( Tr 
~P A  ->  Tr  A )

Proof of Theorem pwtrrVD
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 4246 . . 3  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
2 idn1 28007 . . . . . . . 8  |-  (. Tr  ~P A  ->.  Tr  ~P A ).
3 idn2 28056 . . . . . . . . 9  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  ( z  e.  y  /\  y  e.  A ) ).
4 simpr 448 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  A )  ->  y  e.  A )
53, 4e2 28074 . . . . . . . 8  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  e.  A ).
6 pwtrrVD.1 . . . . . . . . 9  |-  A  e. 
_V
76pwid 3756 . . . . . . . 8  |-  A  e. 
~P A
8 trel 4251 . . . . . . . . 9  |-  ( Tr 
~P A  ->  (
( y  e.  A  /\  A  e.  ~P A )  ->  y  e.  ~P A ) )
98exp3a 426 . . . . . . . 8  |-  ( Tr 
~P A  ->  (
y  e.  A  -> 
( A  e.  ~P A  ->  y  e.  ~P A ) ) )
102, 5, 7, 9e120 28106 . . . . . . 7  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  e.  ~P A ).
11 elpwi 3751 . . . . . . 7  |-  ( y  e.  ~P A  -> 
y  C_  A )
1210, 11e2 28074 . . . . . 6  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  C_  A ).
13 simpl 444 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  y )
143, 13e2 28074 . . . . . 6  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  z  e.  y ).
15 ssel 3286 . . . . . 6  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
1612, 14, 15e22 28114 . . . . 5  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  z  e.  A ).
1716in2 28048 . . . 4  |-  (. Tr  ~P A  ->.  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ).
1817gen12 28061 . . 3  |-  (. Tr  ~P A  ->.  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ).
19 bi2 190 . . 3  |-  ( ( Tr  A  <->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )  -> 
( A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A )  ->  Tr  A ) )
201, 18, 19e01 28134 . 2  |-  (. Tr  ~P A  ->.  Tr  A ).
2120in1 28004 1  |-  ( Tr 
~P A  ->  Tr  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    e. wcel 1717   _Vcvv 2900    C_ wss 3264   ~Pcpw 3743   Tr wtr 4244
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 2369
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 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-in 3271  df-ss 3278  df-pw 3745  df-uni 3959  df-tr 4245  df-vd1 28003  df-vd2 28012
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