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Mirrors > Home > NFE Home > Th. List > caov4 | GIF version |
Description: Rearrange arguments in a commutative, associative operation. (Contributed by set.mm contributors, 26-Aug-1995.) |
Ref | Expression |
---|---|
caopr.1 | ⊢ A ∈ V |
caopr.2 | ⊢ B ∈ V |
caopr.3 | ⊢ C ∈ V |
caopr.com | ⊢ (xFy) = (yFx) |
caopr.ass | ⊢ ((xFy)Fz) = (xF(yFz)) |
caopr.4 | ⊢ D ∈ V |
Ref | Expression |
---|---|
caov4 | ⊢ ((AFB)F(CFD)) = ((AFC)F(BFD)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caopr.2 | . . . 4 ⊢ B ∈ V | |
2 | caopr.3 | . . . 4 ⊢ C ∈ V | |
3 | caopr.4 | . . . 4 ⊢ D ∈ V | |
4 | caopr.com | . . . 4 ⊢ (xFy) = (yFx) | |
5 | caopr.ass | . . . 4 ⊢ ((xFy)Fz) = (xF(yFz)) | |
6 | 1, 2, 3, 4, 5 | caov12 5636 | . . 3 ⊢ (BF(CFD)) = (CF(BFD)) |
7 | 6 | oveq2i 5534 | . 2 ⊢ (AF(BF(CFD))) = (AF(CF(BFD))) |
8 | caopr.1 | . . 3 ⊢ A ∈ V | |
9 | ovex 5551 | . . 3 ⊢ (CFD) ∈ V | |
10 | 8, 1, 9, 5 | caovass 5627 | . 2 ⊢ ((AFB)F(CFD)) = (AF(BF(CFD))) |
11 | ovex 5551 | . . 3 ⊢ (BFD) ∈ V | |
12 | 8, 2, 11, 5 | caovass 5627 | . 2 ⊢ ((AFC)F(BFD)) = (AF(CF(BFD))) |
13 | 7, 10, 12 | 3eqtr4i 2383 | 1 ⊢ ((AFB)F(CFD)) = ((AFC)F(BFD)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∈ wcel 1710 Vcvv 2859 (class class class)co 5525 |
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-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-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 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-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-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-br 4640 df-fv 4795 df-ov 5526 |
This theorem is referenced by: caov42 5641 |
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