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Mirrors > Home > NFE Home > Th. List > vtoclr | GIF version |
Description: Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) |
Ref | Expression |
---|---|
vtoclr.2 | ⊢ ((xRy ∧ yRz) → xRz) |
Ref | Expression |
---|---|
vtoclr | ⊢ ((ARB ∧ BRC) → ARC) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brex 4689 | . . 3 ⊢ (ARB → (A ∈ V ∧ B ∈ V)) | |
2 | brex 4689 | . . . 4 ⊢ (BRC → (B ∈ V ∧ C ∈ V)) | |
3 | 2 | simprd 449 | . . 3 ⊢ (BRC → C ∈ V) |
4 | 1, 3 | anim12i 549 | . 2 ⊢ ((ARB ∧ BRC) → ((A ∈ V ∧ B ∈ V) ∧ C ∈ 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 ∈ V → ((xRy ∧ yRC) → xRC)) ↔ (C ∈ 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 ∈ V → ((ARy ∧ yRC) → ARC)) ↔ (C ∈ 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 ∈ V → ((xRy ∧ yRC) → xRC)) |
21 | 9, 14, 20 | vtocl2g 2918 | . . 3 ⊢ ((A ∈ V ∧ B ∈ V) → (C ∈ V → ((ARB ∧ BRC) → ARC))) |
22 | 21 | imp 418 | . 2 ⊢ (((A ∈ V ∧ B ∈ V) ∧ C ∈ V) → ((ARB ∧ BRC) → ARC)) |
23 | 4, 22 | mpcom 32 | 1 ⊢ ((ARB ∧ BRC) → ARC) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 Vcvv 2859 class class class wbr 4639 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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) |
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