Theorem pwtrrVD 34044

Description: Virtual deduction proof of pwtr 4690; see pwtrVD 34043 for the converse. (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.

Step	Hyp	Ref	Expression
1		dftr2 4534	. . 3 |- ( Tr A <-> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) )
2		idn1 33764	. . . . . . . 8 |- (. Tr ~P A ->. Tr ~P A ).
3		idn2 33812	. . . . . . . . 9 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. ( z e. y /\ y e. A ) ).
4		simpr 459	. . . . . . . . 9 |- ( ( z e. y /\ y e. A ) -> y e. A )
5	3, 4	e2 33830	. . . . . . . 8 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. y e. A ).
6		pwtrrVD.1	. . . . . . . . 9 |- A e. _V
7	6	pwid 4013	. . . . . . . 8 |- A e. ~P A
8		trel 4539	. . . . . . . . 9 |- ( Tr ~P A -> ( ( y e. A /\ A e. ~P A ) -> y e. ~P A ) )
9	8	expd 434	. . . . . . . 8 |- ( Tr ~P A -> ( y e. A -> ( A e. ~P A -> y e. ~P A ) ) )
10	2, 5, 7, 9	e120 33862	. . . . . . 7 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. y e. ~P A ).
11		elpwi 4008	. . . . . . 7 |- ( y e. ~P A -> y C_ A )
12	10, 11	e2 33830	. . . . . 6 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. y C_ A ).
13		simpl 455	. . . . . . 7 |- ( ( z e. y /\ y e. A ) -> z e. y )
14	3, 13	e2 33830	. . . . . 6 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. z e. y ).
15		ssel 3483	. . . . . 6 |- ( y C_ A -> ( z e. y -> z e. A ) )
16	12, 14, 15	e22 33870	. . . . 5 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. z e. A ).
17	16	in2 33804	. . . 4 |- (. Tr ~P A ->. ( ( z e. y /\ y e. A ) -> z e. A ) ).
18	17	gen12 33817	. . 3 |- (. Tr ~P A ->. A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) ).
19		bi2 198	. . 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 ) )
20	1, 18, 19	e01 33890	. 2 |- (. Tr ~P A ->. Tr A ).
21	20	in1 33761	1 |- ( Tr ~P A -> Tr A )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   A.wal 1396   e. wcel 1823   _Vcvv 3106   C_ wss 3461   ~Pcpw 3999   Tr wtr 4532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-in 3468  df-ss 3475  df-pw 4001  df-uni 4236  df-tr 4533  df-vd1 33760  df-vd2 33768

This theorem is referenced by: (None)