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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 2761 | . 2 ⊢ (y = A → ([y / x]B𝑅𝐶 ↔ [A / x]B𝑅𝐶)) | |
2 | csbeq1 2849 | . . 3 ⊢ (y = A → ⦋y / x⦌B = ⦋A / x⦌B) | |
3 | csbeq1 2849 | . . 3 ⊢ (y = A → ⦋y / x⦌𝑅 = ⦋A / x⦌𝑅) | |
4 | csbeq1 2849 | . . 3 ⊢ (y = A → ⦋y / x⦌𝐶 = ⦋A / x⦌𝐶) | |
5 | 2, 3, 4 | breq123d 3769 | . 2 ⊢ (y = A → (⦋y / x⦌B⦋y / x⦌𝑅⦋y / x⦌𝐶 ↔ ⦋A / x⦌B⦋A / x⦌𝑅⦋A / x⦌𝐶)) |
6 | nfcsb1v 2876 | . . . 4 ⊢ Ⅎx⦋y / x⦌B | |
7 | nfcsb1v 2876 | . . . 4 ⊢ Ⅎx⦋y / x⦌𝑅 | |
8 | nfcsb1v 2876 | . . . 4 ⊢ Ⅎx⦋y / x⦌𝐶 | |
9 | 6, 7, 8 | nfbr 3799 | . . 3 ⊢ Ⅎx⦋y / x⦌B⦋y / x⦌𝑅⦋y / x⦌𝐶 |
10 | csbeq1a 2854 | . . . 4 ⊢ (x = y → B = ⦋y / x⦌B) | |
11 | csbeq1a 2854 | . . . 4 ⊢ (x = y → 𝑅 = ⦋y / x⦌𝑅) | |
12 | csbeq1a 2854 | . . . 4 ⊢ (x = y → 𝐶 = ⦋y / x⦌𝐶) | |
13 | 10, 11, 12 | breq123d 3769 | . . 3 ⊢ (x = y → (B𝑅𝐶 ↔ ⦋y / x⦌B⦋y / x⦌𝑅⦋y / x⦌𝐶)) |
14 | 9, 13 | sbie 1671 | . 2 ⊢ ([y / x]B𝑅𝐶 ↔ ⦋y / x⦌B⦋y / x⦌𝑅⦋y / x⦌𝐶) |
15 | 1, 5, 14 | vtoclbg 2608 | 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 98 = wceq 1242 ∈ wcel 1390 [wsb 1642 [wsbc 2758 ⦋csb 2846 class class class wbr 3755 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 |
This theorem is referenced by: sbcbr12g 3805 csbcnvg 4462 sbcfung 4868 csbfv12g 5152 |
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