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Mirrors > Home > MPE Home > Th. List > ecovcom | Structured version Visualization version Unicode version |
Description: Lemma used to transfer a commutative law via an equivalence relation. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.) |
Ref | Expression |
---|---|
ecovcom.1 | |
ecovcom.2 | |
ecovcom.3 | |
ecovcom.4 | |
ecovcom.5 |
Ref | Expression |
---|---|
ecovcom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecovcom.1 | . 2 | |
2 | oveq1 6657 | . . 3 | |
3 | oveq2 6658 | . . 3 | |
4 | 2, 3 | eqeq12d 2637 | . 2 |
5 | oveq2 6658 | . . 3 | |
6 | oveq1 6657 | . . 3 | |
7 | 5, 6 | eqeq12d 2637 | . 2 |
8 | ecovcom.4 | . . . 4 | |
9 | ecovcom.5 | . . . 4 | |
10 | opeq12 4404 | . . . . 5 | |
11 | 10 | eceq1d 7783 | . . . 4 |
12 | 8, 9, 11 | mp2an 708 | . . 3 |
13 | ecovcom.2 | . . 3 | |
14 | ecovcom.3 | . . . 4 | |
15 | 14 | ancoms 469 | . . 3 |
16 | 12, 13, 15 | 3eqtr4a 2682 | . 2 |
17 | 1, 4, 7, 16 | 2ecoptocl 7838 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cop 4183 cxp 5112 (class class class)co 6650 cec 7740 cqs 7741 |
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 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
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-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fv 5896 df-ov 6653 df-ec 7744 df-qs 7748 |
This theorem is referenced by: addcomsr 9908 mulcomsr 9910 axmulcom 9976 |
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