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Mirrors > Home > NFE Home > Th. List > caovass | GIF version |
Description: Convert an operation associative law to class notation. (Contributed by set.mm contributors, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.) |
Ref | Expression |
---|---|
caovass.1 | ⊢ A ∈ V |
caovass.2 | ⊢ B ∈ V |
caovass.3 | ⊢ C ∈ V |
caovass.4 | ⊢ ((xFy)Fz) = (xF(yFz)) |
Ref | Expression |
---|---|
caovass | ⊢ ((AFB)FC) = (AF(BFC)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovass.1 | . 2 ⊢ A ∈ V | |
2 | caovass.2 | . 2 ⊢ B ∈ V | |
3 | caovass.3 | . 2 ⊢ C ∈ V | |
4 | tru 1321 | . . 3 ⊢ ⊤ | |
5 | caovass.4 | . . . . 5 ⊢ ((xFy)Fz) = (xF(yFz)) | |
6 | 5 | a1i 10 | . . . 4 ⊢ (( ⊤ ∧ (x ∈ V ∧ y ∈ V ∧ z ∈ V)) → ((xFy)Fz) = (xF(yFz))) |
7 | 6 | caovassg 5626 | . . 3 ⊢ (( ⊤ ∧ (A ∈ V ∧ B ∈ V ∧ C ∈ V)) → ((AFB)FC) = (AF(BFC))) |
8 | 4, 7 | mpan 651 | . 2 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → ((AFB)FC) = (AF(BFC))) |
9 | 1, 2, 3, 8 | mp3an 1277 | 1 ⊢ ((AFB)FC) = (AF(BFC)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∧ w3a 934 ⊤ wtru 1316 = wceq 1642 ∈ wcel 1710 Vcvv 2859 (class class class)co 5525 |
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-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-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: caov32 5635 caov12 5636 caov31 5637 caov13 5638 caov4 5639 caov411 5640 caovdilem 5643 caovmo 5645 |
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