Theorem trd 5922

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

Hypotheses

Ref	Expression
trd.1	⊢ (φ → R Trans A)
trd.2	⊢ (φ → X ∈ A)
trd.3	⊢ (φ → Y ∈ A)
trd.4	⊢ (φ → Z ∈ A)
trd.5	⊢ (φ → XRY)
trd.6	⊢ (φ → YRZ)

Assertion

Ref	Expression
trd	⊢ (φ → 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 ⊢ (φ → XRY)
2		trd.6	. 2 ⊢ (φ → YRZ)
3		trd.1	. . . 4 ⊢ (φ → R Trans A)
4		brex 4690	. . . . . 6 ⊢ (R Trans A → (R ∈ V ∧ A ∈ 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 → (∀z ∈ a ((xry ∧ yrz) → xrz) ↔ ∀z ∈ a ((xRy ∧ yRz) → xRz)))
11	10	2ralbidv 2657	. . . . . . 7 ⊢ (r = R → (∀x ∈ a ∀y ∈ a ∀z ∈ a ((xry ∧ yrz) → xrz) ↔ ∀x ∈ a ∀y ∈ a ∀z ∈ a ((xRy ∧ yRz) → xRz)))
12		raleq 2808	. . . . . . . . 9 ⊢ (a = A → (∀z ∈ a ((xRy ∧ yRz) → xRz) ↔ ∀z ∈ A ((xRy ∧ yRz) → xRz)))
13	12	raleqbi1dv 2816	. . . . . . . 8 ⊢ (a = A → (∀y ∈ a ∀z ∈ a ((xRy ∧ yRz) → xRz) ↔ ∀y ∈ A ∀z ∈ A ((xRy ∧ yRz) → xRz)))
14	13	raleqbi1dv 2816	. . . . . . 7 ⊢ (a = A → (∀x ∈ a ∀y ∈ a ∀z ∈ a ((xRy ∧ yRz) → xRz) ↔ ∀x ∈ A ∀y ∈ A ∀z ∈ A ((xRy ∧ yRz) → xRz)))
15		df-trans 5900	. . . . . . 7 ⊢ Trans = {⟨r, a⟩ ∣ ∀x ∈ a ∀y ∈ a ∀z ∈ a ((xry ∧ yrz) → xrz)}
16	11, 14, 15	brabg 4707	. . . . . 6 ⊢ ((R ∈ V ∧ A ∈ V) → (R Trans A ↔ ∀x ∈ A ∀y ∈ A ∀z ∈ A ((xRy ∧ yRz) → xRz)))
17	4, 16	syl 15	. . . . 5 ⊢ (R Trans A → (R Trans A ↔ ∀x ∈ A ∀y ∈ A ∀z ∈ A ((xRy ∧ yRz) → xRz)))
18	17	ibi 232	. . . 4 ⊢ (R Trans A → ∀x ∈ A ∀y ∈ A ∀z ∈ A ((xRy ∧ yRz) → xRz))
19	3, 18	syl 15	. . 3 ⊢ (φ → ∀x ∈ A ∀y ∈ A ∀z ∈ A ((xRy ∧ yRz) → xRz))
20		trd.2	. . . 4 ⊢ (φ → X ∈ A)
21		trd.3	. . . 4 ⊢ (φ → Y ∈ A)
22		trd.4	. . . 4 ⊢ (φ → Z ∈ 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 ∈ A ∧ Y ∈ A ∧ Z ∈ A) → (∀x ∈ A ∀y ∈ A ∀z ∈ A ((xRy ∧ yRz) → xRz) → ((XRY ∧ YRZ) → XRZ)))
36	20, 21, 22, 35	syl3anc 1182	. . 3 ⊢ (φ → (∀x ∈ A ∀y ∈ A ∀z ∈ A ((xRy ∧ yRz) → xRz) → ((XRY ∧ YRZ) → XRZ)))
37	19, 36	mpd 14	. 2 ⊢ (φ → ((XRY ∧ YRZ) → XRZ))
38	1, 2, 37	mp2and 660	1 ⊢ (φ → XRZ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀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