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Mirrors > Home > ILE Home > Th. List > csbriotag | GIF version |
Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) |
Ref | Expression |
---|---|
csbriotag | ⊢ (A ∈ 𝑉 → ⦋A / x⦌(℩y ∈ B φ) = (℩y ∈ B [A / x]φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 2849 | . . 3 ⊢ (z = A → ⦋z / x⦌(℩y ∈ B φ) = ⦋A / x⦌(℩y ∈ B φ)) | |
2 | dfsbcq2 2761 | . . . 4 ⊢ (z = A → ([z / x]φ ↔ [A / x]φ)) | |
3 | 2 | riotabidv 5413 | . . 3 ⊢ (z = A → (℩y ∈ B [z / x]φ) = (℩y ∈ B [A / x]φ)) |
4 | 1, 3 | eqeq12d 2051 | . 2 ⊢ (z = A → (⦋z / x⦌(℩y ∈ B φ) = (℩y ∈ B [z / x]φ) ↔ ⦋A / x⦌(℩y ∈ B φ) = (℩y ∈ B [A / x]φ))) |
5 | vex 2554 | . . 3 ⊢ z ∈ V | |
6 | nfs1v 1812 | . . . 4 ⊢ Ⅎx[z / x]φ | |
7 | nfcv 2175 | . . . 4 ⊢ ℲxB | |
8 | 6, 7 | nfriota 5420 | . . 3 ⊢ Ⅎx(℩y ∈ B [z / x]φ) |
9 | sbequ12 1651 | . . . 4 ⊢ (x = z → (φ ↔ [z / x]φ)) | |
10 | 9 | riotabidv 5413 | . . 3 ⊢ (x = z → (℩y ∈ B φ) = (℩y ∈ B [z / x]φ)) |
11 | 5, 8, 10 | csbief 2885 | . 2 ⊢ ⦋z / x⦌(℩y ∈ B φ) = (℩y ∈ B [z / x]φ) |
12 | 4, 11 | vtoclg 2607 | 1 ⊢ (A ∈ 𝑉 → ⦋A / x⦌(℩y ∈ B φ) = (℩y ∈ B [A / x]φ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 [wsb 1642 [wsbc 2758 ⦋csb 2846 ℩crio 5410 |
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-sn 3373 df-uni 3572 df-iota 4810 df-riota 5411 |
This theorem is referenced by: (None) |
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