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Mirrors > Home > NFE Home > Th. List > addceq12i | GIF version |
Description: Equality inference for cardinal addition. (Contributed by SF, 3-Feb-2015.) |
Ref | Expression |
---|---|
addceqi.1 | ⊢ A = B |
addceqi.2 | ⊢ C = D |
Ref | Expression |
---|---|
addceq12i | ⊢ (A +c C) = (B +c D) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addceqi.1 | . 2 ⊢ A = B | |
2 | addceqi.2 | . 2 ⊢ C = D | |
3 | addceq12 4385 | . 2 ⊢ ((A = B ∧ C = D) → (A +c C) = (B +c D)) | |
4 | 1, 2, 3 | mp2an 653 | 1 ⊢ (A +c C) = (B +c D) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 +c cplc 4375 |
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-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-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: 1p1e2c 6155 2p1e3c 6156 tc2c 6166 ce0addcnnul 6179 addcdi 6250 nchoicelem2 6290 |
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