Theorem sstrALT2VD 39069

Description: Virtual deduction proof of sstrALT2 39070. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sstrALT2VD	|- ( ( A C_ B /\ B C_ C ) -> A C_ C )

Proof of Theorem sstrALT2VD

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3591	. . 3 |- ( A C_ C <-> A. x ( x e. A -> x e. C ) )
2		idn1 38790	. . . . . . 7 |- (. ( A C_ B /\ B C_ C ) ->. ( A C_ B /\ B C_ C ) ).
3		simpr 477	. . . . . . 7 |- ( ( A C_ B /\ B C_ C ) -> B C_ C )
4	2, 3	e1a 38852	. . . . . 6 |- (. ( A C_ B /\ B C_ C ) ->. B C_ C ).
5		simpl 473	. . . . . . . 8 |- ( ( A C_ B /\ B C_ C ) -> A C_ B )
6	2, 5	e1a 38852	. . . . . . 7 |- (. ( A C_ B /\ B C_ C ) ->. A C_ B ).
7		idn2 38838	. . . . . . 7 |- (. ( A C_ B /\ B C_ C ) ,. x e. A ->. x e. A ).
8		ssel2 3598	. . . . . . 7 |- ( ( A C_ B /\ x e. A ) -> x e. B )
9	6, 7, 8	e12an 38952	. . . . . 6 |- (. ( A C_ B /\ B C_ C ) ,. x e. A ->. x e. B ).
10		ssel2 3598	. . . . . 6 |- ( ( B C_ C /\ x e. B ) -> x e. C )
11	4, 9, 10	e12an 38952	. . . . 5 |- (. ( A C_ B /\ B C_ C ) ,. x e. A ->. x e. C ).
12	11	in2 38830	. . . 4 |- (. ( A C_ B /\ B C_ C ) ->. ( x e. A -> x e. C ) ).
13	12	gen11 38841	. . 3 |- (. ( A C_ B /\ B C_ C ) ->. A. x ( x e. A -> x e. C ) ).
14		biimpr 210	. . 3 |- ( ( A C_ C <-> A. x ( x e. A -> x e. C ) ) -> ( A. x ( x e. A -> x e. C ) -> A C_ C ) )
15	1, 13, 14	e01 38916	. 2 |- (. ( A C_ B /\ B C_ C ) ->. A C_ C ).
16	15	in1 38787	1 |- ( ( A C_ B /\ B C_ C ) -> A C_ C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-vd1 38786  df-vd2 38794

This theorem is referenced by: (None)