![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > seqeq123d | GIF version |
Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.) |
Ref | Expression |
---|---|
seqeq123d.1 | ⊢ (𝜑 → 𝑀 = 𝑁) |
seqeq123d.2 | ⊢ (𝜑 → + = 𝑄) |
seqeq123d.3 | ⊢ (𝜑 → 𝐹 = 𝐺) |
Ref | Expression |
---|---|
seqeq123d | ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqeq123d.1 | . . 3 ⊢ (𝜑 → 𝑀 = 𝑁) | |
2 | 1 | seqeq1d 10117 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹)) |
3 | seqeq123d.2 | . . 3 ⊢ (𝜑 → + = 𝑄) | |
4 | 3 | seqeq2d 10118 | . 2 ⊢ (𝜑 → seq𝑁( + , 𝐹) = seq𝑁(𝑄, 𝐹)) |
5 | seqeq123d.3 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
6 | 5 | seqeq3d 10119 | . 2 ⊢ (𝜑 → seq𝑁(𝑄, 𝐹) = seq𝑁(𝑄, 𝐺)) |
7 | 2, 4, 6 | 3eqtrd 2151 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1314 seqcseq 10111 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-mpt 3951 df-cnv 4507 df-dm 4509 df-rn 4510 df-res 4511 df-iota 5046 df-fv 5089 df-ov 5731 df-oprab 5732 df-mpo 5733 df-recs 6156 df-frec 6242 df-seqfrec 10112 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |