Theorem pwtrrVD 37260

Description: Virtual deduction proof of pwtr 4666; see pwtrVD 37259 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 4512	. . 3 |- ( Tr A <-> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) )
2		idn1 36986	. . . . . . . 8 |- (. Tr ~P A ->. Tr ~P A ).
3		idn2 37034	. . . . . . . . 9 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. ( z e. y /\ y e. A ) ).
4		simpr 467	. . . . . . . . 9 |- ( ( z e. y /\ y e. A ) -> y e. A )
5	3, 4	e2 37052	. . . . . . . 8 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. y e. A ).
6		pwtrrVD.1	. . . . . . . . 9 |- A e. _V
7	6	pwid 3976	. . . . . . . 8 |- A e. ~P A
8		trel 4517	. . . . . . . . 9 |- ( Tr ~P A -> ( ( y e. A /\ A e. ~P A ) -> y e. ~P A ) )
9	8	expd 442	. . . . . . . 8 |- ( Tr ~P A -> ( y e. A -> ( A e. ~P A -> y e. ~P A ) ) )
10	2, 5, 7, 9	e120 37084	. . . . . . 7 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. y e. ~P A ).
11		elpwi 3971	. . . . . . 7 |- ( y e. ~P A -> y C_ A )
12	10, 11	e2 37052	. . . . . 6 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. y C_ A ).
13		simpl 463	. . . . . . 7 |- ( ( z e. y /\ y e. A ) -> z e. y )
14	3, 13	e2 37052	. . . . . 6 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. z e. y ).
15		ssel 3437	. . . . . 6 |- ( y C_ A -> ( z e. y -> z e. A ) )
16	12, 14, 15	e22 37092	. . . . 5 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. z e. A ).
17	16	in2 37026	. . . 4 |- (. Tr ~P A ->. ( ( z e. y /\ y e. A ) -> z e. A ) ).
18	17	gen12 37039	. . 3 |- (. Tr ~P A ->. A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) ).
19		biimpr 203	. . 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 37112	. 2 |- (. Tr ~P A ->. Tr A ).
21	20	in1 36983	1 |- ( Tr ~P A -> Tr A )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   A.wal 1452   e. wcel 1897   _Vcvv 3056   C_ wss 3415   ~Pcpw 3962   Tr wtr 4510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-v 3058  df-in 3422  df-ss 3429  df-pw 3964  df-uni 4212  df-tr 4511  df-vd1 36982  df-vd2 36990

This theorem is referenced by: (None)