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Mirrors > Home > ILE Home > Th. List > sbcbrg | GIF version |
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
sbcbrg | ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 2885 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵𝑅𝐶 ↔ [𝐴 / 𝑥]𝐵𝑅𝐶)) | |
2 | csbeq1 2978 | . . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
3 | csbeq1 2978 | . . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌𝑅) | |
4 | csbeq1 2978 | . . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶) | |
5 | 2, 3, 4 | breq123d 3913 | . 2 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶)) |
6 | nfcsb1v 3005 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
7 | nfcsb1v 3005 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅 | |
8 | nfcsb1v 3005 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
9 | 6, 7, 8 | nfbr 3944 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶 |
10 | csbeq1a 2983 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
11 | csbeq1a 2983 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝑅 = ⦋𝑦 / 𝑥⦌𝑅) | |
12 | csbeq1a 2983 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
13 | 10, 11, 12 | breq123d 3913 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐵𝑅𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶)) |
14 | 9, 13 | sbie 1749 | . 2 ⊢ ([𝑦 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶) |
15 | 1, 5, 14 | vtoclbg 2721 | 1 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∈ wcel 1465 [wsb 1720 [wsbc 2882 ⦋csb 2975 class class class wbr 3899 |
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-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 |
This theorem is referenced by: sbcbr12g 3953 csbcnvg 4693 sbcfung 5117 csbfv12g 5425 |
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