Theorem sstrALT2VD 16658

Description: Virtual deduction proof of sstrALT2 16659.

Assertion

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

Proof of Theorem sstrALT2VD

Step	Hyp	Ref	Expression
1		dfss2 2610	. . 3 |- (A C_ C <-> A.x(x e. A -> x e. C))
2		idn1 16484	. . . . . . 7 |- . (A C_ B /\ B C_ C)   ⊢   (A C_ B /\ B C_ C) .
3		simpr 350	. . . . . . 7 |- ((A C_ B /\ B C_ C) -> B C_ C)
4	2, 3	e1_ 16518	. . . . . 6 |- . (A C_ B /\ B C_ C)   ⊢   B C_ C .
5		simpl 346	. . . . . . . 8 |- ((A C_ B /\ B C_ C) -> A C_ B)
6	2, 5	e1_ 16518	. . . . . . 7 |- . (A C_ B /\ B C_ C)   ⊢   A C_ B .
7		idn2 16509	. . . . . . 7 |- . (A C_ B /\ B C_ C), x e. A   ⊢   x e. A .
8		ssel2 2616	. . . . . . 7 |- ((A C_ B /\ x e. A) -> x e. B)
9	6, 7, 8	e12an 16594	. . . . . 6 |- . (A C_ B /\ B C_ C), x e. A   ⊢   x e. B .
10		ssel2 2616	. . . . . 6 |- ((B C_ C /\ x e. B) -> x e. C)
11	4, 9, 10	e12an 16594	. . . . 5 |- . (A C_ B /\ B C_ C), x e. A   ⊢   x e. C .
12	11	in2 16506	. . . 4 |- . (A C_ B /\ B C_ C)   ⊢   (x e. A -> x e. C) .
13	12	gen11 16511	. . 3 |- . (A C_ B /\ B C_ C)   ⊢   A.x(x e. A -> x e. C) .
14		bi2 166	. . 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 16581	. 2 |- . (A C_ B /\ B C_ C)   ⊢   A C_ C .
16	15	in1 16481	1 |- ((A C_ B /\ B C_ C) -> A C_ C)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300   C_ wss 2593

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-in 2603  df-ss 2605  df-vd1 16480  df-vd2 16489

Copyright terms: Public domain