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Mirrors > Home > MPE Home > Th. List > caov31 | Structured version Visualization version Unicode version |
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) |
Ref | Expression |
---|---|
caov.1 | |
caov.2 | |
caov.3 | |
caov.com | |
caov.ass |
Ref | Expression |
---|---|
caov31 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caov.1 | . . . 4 | |
2 | caov.3 | . . . 4 | |
3 | caov.2 | . . . 4 | |
4 | caov.ass | . . . 4 | |
5 | 1, 2, 3, 4 | caovass 6834 | . . 3 |
6 | caov.com | . . . 4 | |
7 | 1, 2, 3, 6, 4 | caov12 6862 | . . 3 |
8 | 5, 7 | eqtri 2644 | . 2 |
9 | 1, 3, 2, 6, 4 | caov32 6861 | . 2 |
10 | 2, 1, 3, 6, 4 | caov32 6861 | . . 3 |
11 | 2, 1, 3, 4 | caovass 6834 | . . 3 |
12 | 10, 11 | eqtr3i 2646 | . 2 |
13 | 8, 9, 12 | 3eqtr4i 2654 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 cvv 3200 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: caov13 6864 caov411 6866 |
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