Theorem pwtrrVD 31861

Description: Virtual deduction proof of pwtr 4643; see pwtrVD 31860 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 4485	. . 3 |- ( Tr A <-> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) )
2		idn1 31587	. . . . . . . 8 |- (. Tr ~P A ->. Tr ~P A ).
3		idn2 31635	. . . . . . . . 9 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. ( z e. y /\ y e. A ) ).
4		simpr 461	. . . . . . . . 9 |- ( ( z e. y /\ y e. A ) -> y e. A )
5	3, 4	e2 31653	. . . . . . . 8 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. y e. A ).
6		pwtrrVD.1	. . . . . . . . 9 |- A e. _V
7	6	pwid 3972	. . . . . . . 8 |- A e. ~P A
8		trel 4490	. . . . . . . . 9 |- ( Tr ~P A -> ( ( y e. A /\ A e. ~P A ) -> y e. ~P A ) )
9	8	expd 436	. . . . . . . 8 |- ( Tr ~P A -> ( y e. A -> ( A e. ~P A -> y e. ~P A ) ) )
10	2, 5, 7, 9	e120 31685	. . . . . . 7 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. y e. ~P A ).
11		elpwi 3967	. . . . . . 7 |- ( y e. ~P A -> y C_ A )
12	10, 11	e2 31653	. . . . . 6 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. y C_ A ).
13		simpl 457	. . . . . . 7 |- ( ( z e. y /\ y e. A ) -> z e. y )
14	3, 13	e2 31653	. . . . . 6 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. z e. y ).
15		ssel 3448	. . . . . 6 |- ( y C_ A -> ( z e. y -> z e. A ) )
16	12, 14, 15	e22 31693	. . . . 5 |- (. Tr ~P A ,. ( z e. y /\ y e. A ) ->. z e. A ).
17	16	in2 31627	. . . 4 |- (. Tr ~P A ->. ( ( z e. y /\ y e. A ) -> z e. A ) ).
18	17	gen12 31640	. . 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 31713	. 2 |- (. Tr ~P A ->. Tr A ).
21	20	in1 31584	1 |- ( Tr ~P A -> Tr A )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1368   e. wcel 1758   _Vcvv 3068   C_ wss 3426   ~Pcpw 3958   Tr wtr 4483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3070  df-in 3433  df-ss 3440  df-pw 3960  df-uni 4190  df-tr 4484  df-vd1 31583  df-vd2 31591

This theorem is referenced by: (None)