Theorem vtoclr 4816

Description: Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.)

Hypothesis

Ref	Expression
vtoclr.2	|- ((xRy /\ yRz) -> xRz)

Assertion

Ref	Expression
vtoclr	|- ((ARB /\ BRC) -> ARC)

Distinct variable groups:   x,y,A   y,B   x,z,C,y   x,R,y,z

Allowed substitution hints:   A(z)   B(x,z)

Proof of Theorem vtoclr

Step	Hyp	Ref	Expression
1		brex 4689	. . 3 |- (ARB -> (A e. _V /\ B e. _V))
2		brex 4689	. . . 4 |- (BRC -> (B e. _V /\ C e. _V))
3	2	simprd 449	. . 3 |- (BRC -> C e. _V)
4	1, 3	anim12i 549	. 2 |- ((ARB /\ BRC) -> ((A e. _V /\ B e. _V) /\ C e. _V))
5		breq1 4642	. . . . . . 7 |- (x = A -> (xRy <-> ARy))
6	5	anbi1d 685	. . . . . 6 |- (x = A -> ((xRy /\ yRC) <-> (ARy /\ yRC)))
7		breq1 4642	. . . . . 6 |- (x = A -> (xRC <-> ARC))
8	6, 7	imbi12d 311	. . . . 5 |- (x = A -> (((xRy /\ yRC) -> xRC) <-> ((ARy /\ yRC) -> ARC)))
9	8	imbi2d 307	. . . 4 |- (x = A -> ((C e. _V -> ((xRy /\ yRC) -> xRC)) <-> (C e. _V -> ((ARy /\ yRC) -> ARC))))
10		breq2 4643	. . . . . . 7 |- (y = B -> (ARy <-> ARB))
11		breq1 4642	. . . . . . 7 |- (y = B -> (yRC <-> BRC))
12	10, 11	anbi12d 691	. . . . . 6 |- (y = B -> ((ARy /\ yRC) <-> (ARB /\ BRC)))
13	12	imbi1d 308	. . . . 5 |- (y = B -> (((ARy /\ yRC) -> ARC) <-> ((ARB /\ BRC) -> ARC)))
14	13	imbi2d 307	. . . 4 |- (y = B -> ((C e. _V -> ((ARy /\ yRC) -> ARC)) <-> (C e. _V -> ((ARB /\ BRC) -> ARC))))
15		breq2 4643	. . . . . . 7 |- (z = C -> (yRz <-> yRC))
16	15	anbi2d 684	. . . . . 6 |- (z = C -> ((xRy /\ yRz) <-> (xRy /\ yRC)))
17		breq2 4643	. . . . . 6 |- (z = C -> (xRz <-> xRC))
18	16, 17	imbi12d 311	. . . . 5 |- (z = C -> (((xRy /\ yRz) -> xRz) <-> ((xRy /\ yRC) -> xRC)))
19		vtoclr.2	. . . . 5 |- ((xRy /\ yRz) -> xRz)
20	18, 19	vtoclg 2914	. . . 4 |- (C e. _V -> ((xRy /\ yRC) -> xRC))
21	9, 14, 20	vtocl2g 2918	. . 3 |- ((A e. _V /\ B e. _V) -> (C e. _V -> ((ARB /\ BRC) -> ARC)))
22	21	imp 418	. 2 |- (((A e. _V /\ B e. _V) /\ C e. _V) -> ((ARB /\ BRC) -> ARC))
23	4, 22	mpcom 32	1 |- ((ARB /\ BRC) -> ARC)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  _Vcvv 2859   class class class wbr 4639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-br 4640

This theorem is referenced by: (None)