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Mirrors > Home > NFE Home > Th. List > addc32 | GIF version |
Description: Swap arguments two and three in cardinal addition. (Contributed by SF, 22-Jan-2015.) |
Ref | Expression |
---|---|
addc32 | ⊢ ((A +c B) +c C) = ((A +c C) +c B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addccom 4406 | . . 3 ⊢ (B +c C) = (C +c B) | |
2 | 1 | addceq2i 4387 | . 2 ⊢ (A +c (B +c C)) = (A +c (C +c B)) |
3 | addcass 4415 | . 2 ⊢ ((A +c B) +c C) = (A +c (B +c C)) | |
4 | addcass 4415 | . 2 ⊢ ((A +c C) +c B) = (A +c (C +c B)) | |
5 | 2, 3, 4 | 3eqtr4i 2383 | 1 ⊢ ((A +c B) +c C) = ((A +c C) +c B) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 +c cplc 4375 |
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-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-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-pw 3724 df-sn 3741 df-pr 3742 df-opk 4058 df-1c 4136 df-pw1 4137 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-sik 4192 df-ssetk 4193 df-addc 4378 |
This theorem is referenced by: addc4 4417 addc6 4418 preaddccan2 4455 ltfintri 4466 sucoddeven 4511 evenodddisj 4516 leaddc1 6214 addceq0 6219 addccan2nc 6265 nncdiv3 6277 nnc3n3p1 6278 nnc3n3p2 6279 nchoicelem2 6290 nchoicelem17 6305 |
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