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Mirrors > Home > ILE Home > Th. List > seqeq1d | GIF version |
Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.) |
Ref | Expression |
---|---|
seqeqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
seqeq1d | ⊢ (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqeqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | seqeq1 10189 | . 2 ⊢ (𝐴 = 𝐵 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 seqcseq 10186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-iota 5058 df-fv 5101 df-oprab 5746 df-mpo 5747 df-recs 6170 df-frec 6256 df-seqfrec 10187 |
This theorem is referenced by: seqeq123d 10195 seq3f1olemqsum 10241 bcval5 10477 bcn2 10478 seq3shft 10578 iserex 11076 iser3shft 11083 isumsplit 11228 eftlub 11323 |
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