Theorem trd 5922

Description: Transitivity law in natural deduction form. (Contributed by SF, 20-Feb-2015.)

Hypotheses

Ref	Expression
trd.1	|- (ph -> R Trans A)
trd.2	|- (ph -> X e. A)
trd.3	|- (ph -> Y e. A)
trd.4	|- (ph -> Z e. A)
trd.5	|- (ph -> XRY)
trd.6	|- (ph -> YRZ)

Assertion

Ref	Expression
trd	|- (ph -> XRZ)

Proof of Theorem trd

Dummy variables a r x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		trd.5	. 2 |- (ph -> XRY)
2		trd.6	. 2 |- (ph -> YRZ)
3		trd.1	. . . 4 |- (ph -> R Trans A)
4		brex 4690	. . . . . 6 |- (R Trans A -> (R e. _V /\ A e. _V))
5		breq 4642	. . . . . . . . . . 11 |- (r = R -> (xry <-> xRy))
6		breq 4642	. . . . . . . . . . 11 |- (r = R -> (yrz <-> yRz))
7	5, 6	anbi12d 691	. . . . . . . . . 10 |- (r = R -> ((xry /\ yrz) <-> (xRy /\ yRz)))
8		breq 4642	. . . . . . . . . 10 |- (r = R -> (xrz <-> xRz))
9	7, 8	imbi12d 311	. . . . . . . . 9 |- (r = R -> (((xry /\ yrz) -> xrz) <-> ((xRy /\ yRz) -> xRz)))
10	9	ralbidv 2635	. . . . . . . 8 |- (r = R -> (A.z e. a ((xry /\ yrz) -> xrz) <-> A.z e. a ((xRy /\ yRz) -> xRz)))
11	10	2ralbidv 2657	. . . . . . 7 |- (r = R -> (A.x e. a A.y e. a A.z e. a ((xry /\ yrz) -> xrz) <-> A.x e. a A.y e. a A.z e. a ((xRy /\ yRz) -> xRz)))
12		raleq 2808	. . . . . . . . 9 |- (a = A -> (A.z e. a ((xRy /\ yRz) -> xRz) <-> A.z e. A ((xRy /\ yRz) -> xRz)))
13	12	raleqbi1dv 2816	. . . . . . . 8 |- (a = A -> (A.y e. a A.z e. a ((xRy /\ yRz) -> xRz) <-> A.y e. A A.z e. A ((xRy /\ yRz) -> xRz)))
14	13	raleqbi1dv 2816	. . . . . . 7 |- (a = A -> (A.x e. a A.y e. a A.z e. a ((xRy /\ yRz) -> xRz) <-> A.x e. A A.y e. A A.z e. A ((xRy /\ yRz) -> xRz)))
15		df-trans 5900	. . . . . . 7 |- Trans = {<.r, a>. | A.x e. a A.y e. a A.z e. a ((xry /\ yrz) -> xrz)}
16	11, 14, 15	brabg 4707	. . . . . 6 |- ((R e. _V /\ A e. _V) -> (R Trans A <-> A.x e. A A.y e. A A.z e. A ((xRy /\ yRz) -> xRz)))
17	4, 16	syl 15	. . . . 5 |- (R Trans A -> (R Trans A <-> A.x e. A A.y e. A A.z e. A ((xRy /\ yRz) -> xRz)))
18	17	ibi 232	. . . 4 |- (R Trans A -> A.x e. A A.y e. A A.z e. A ((xRy /\ yRz) -> xRz))
19	3, 18	syl 15	. . 3 |- (ph -> A.x e. A A.y e. A A.z e. A ((xRy /\ yRz) -> xRz))
20		trd.2	. . . 4 |- (ph -> X e. A)
21		trd.3	. . . 4 |- (ph -> Y e. A)
22		trd.4	. . . 4 |- (ph -> Z e. A)
23		breq1 4643	. . . . . . 7 |- (x = X -> (xRy <-> XRy))
24	23	anbi1d 685	. . . . . 6 |- (x = X -> ((xRy /\ yRz) <-> (XRy /\ yRz)))
25		breq1 4643	. . . . . 6 |- (x = X -> (xRz <-> XRz))
26	24, 25	imbi12d 311	. . . . 5 |- (x = X -> (((xRy /\ yRz) -> xRz) <-> ((XRy /\ yRz) -> XRz)))
27		breq2 4644	. . . . . . 7 |- (y = Y -> (XRy <-> XRY))
28		breq1 4643	. . . . . . 7 |- (y = Y -> (yRz <-> YRz))
29	27, 28	anbi12d 691	. . . . . 6 |- (y = Y -> ((XRy /\ yRz) <-> (XRY /\ YRz)))
30	29	imbi1d 308	. . . . 5 |- (y = Y -> (((XRy /\ yRz) -> XRz) <-> ((XRY /\ YRz) -> XRz)))
31		breq2 4644	. . . . . . 7 |- (z = Z -> (YRz <-> YRZ))
32	31	anbi2d 684	. . . . . 6 |- (z = Z -> ((XRY /\ YRz) <-> (XRY /\ YRZ)))
33		breq2 4644	. . . . . 6 |- (z = Z -> (XRz <-> XRZ))
34	32, 33	imbi12d 311	. . . . 5 |- (z = Z -> (((XRY /\ YRz) -> XRz) <-> ((XRY /\ YRZ) -> XRZ)))
35	26, 30, 34	rspc3v 2965	. . . 4 |- ((X e. A /\ Y e. A /\ Z e. A) -> (A.x e. A A.y e. A A.z e. A ((xRy /\ yRz) -> xRz) -> ((XRY /\ YRZ) -> XRZ)))
36	20, 21, 22, 35	syl3anc 1182	. . 3 |- (ph -> (A.x e. A A.y e. A A.z e. A ((xRy /\ yRz) -> xRz) -> ((XRY /\ YRZ) -> XRZ)))
37	19, 36	mpd 14	. 2 |- (ph -> ((XRY /\ YRZ) -> XRZ))
38	1, 2, 37	mp2and 660	1 |- (ph -> XRZ)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2615  _Vcvv 2860   class class class wbr 4640   Trans ctrans 5889

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-trans 5900

This theorem is referenced by:  ertr  5955  ertrd  5956