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Mirrors > Home > ILE Home > Th. List > csbov2g | GIF version |
Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
Ref | Expression |
---|---|
csbov2g | ⊢ (A ∈ 𝑉 → ⦋A / x⦌(B𝐹𝐶) = (B𝐹⦋A / x⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbov12g 5486 | . 2 ⊢ (A ∈ 𝑉 → ⦋A / x⦌(B𝐹𝐶) = (⦋A / x⦌B𝐹⦋A / x⦌𝐶)) | |
2 | csbconstg 2858 | . . 3 ⊢ (A ∈ 𝑉 → ⦋A / x⦌B = B) | |
3 | 2 | oveq1d 5470 | . 2 ⊢ (A ∈ 𝑉 → (⦋A / x⦌B𝐹⦋A / x⦌𝐶) = (B𝐹⦋A / x⦌𝐶)) |
4 | 1, 3 | eqtrd 2069 | 1 ⊢ (A ∈ 𝑉 → ⦋A / x⦌(B𝐹𝐶) = (B𝐹⦋A / x⦌𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 ⦋csb 2846 (class class class)co 5455 |
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-rex 2306 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-iota 4810 df-fv 4853 df-ov 5458 |
This theorem is referenced by: csbnegg 7006 |
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