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Mirrors > Home > ILE Home > Th. List > csbriotag | Structured version 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 2831 | . . 3 ⊢ (z = A → ⦋z / x⦌(℩y ∈ B φ) = ⦋A / x⦌(℩y ∈ B φ)) | |
2 | dfsbcq2 2743 | . . . 4 ⊢ (z = A → ([z / x]φ ↔ [A / x]φ)) | |
3 | 2 | riotabidv 5393 | . . 3 ⊢ (z = A → (℩y ∈ B [z / x]φ) = (℩y ∈ B [A / x]φ)) |
4 | 1, 3 | eqeq12d 2037 | . 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 2537 | . . 3 ⊢ z ∈ V | |
6 | nfs1v 1798 | . . . 4 ⊢ Ⅎx[z / x]φ | |
7 | nfcv 2161 | . . . 4 ⊢ ℲxB | |
8 | 6, 7 | nfriota 5399 | . . 3 ⊢ Ⅎx(℩y ∈ B [z / x]φ) |
9 | sbequ12 1637 | . . . 4 ⊢ (x = z → (φ ↔ [z / x]φ)) | |
10 | 9 | riotabidv 5393 | . . 3 ⊢ (x = z → (℩y ∈ B φ) = (℩y ∈ B [z / x]φ)) |
11 | 5, 8, 10 | csbief 2867 | . 2 ⊢ ⦋z / x⦌(℩y ∈ B φ) = (℩y ∈ B [z / x]φ) |
12 | 4, 11 | vtoclg 2589 | 1 ⊢ (A ∈ 𝑉 → ⦋A / x⦌(℩y ∈ B φ) = (℩y ∈ B [A / x]φ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1228 ∈ wcel 1375 [wsb 1628 [wsbc 2740 ⦋csb 2828 ℩crio 5390 |
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 617 ax-5 1316 ax-7 1317 ax-gen 1318 ax-ie1 1364 ax-ie2 1365 ax-8 1377 ax-10 1378 ax-11 1379 ax-i12 1380 ax-bnd 1381 ax-4 1382 ax-17 1401 ax-i9 1405 ax-ial 1410 ax-i5r 1411 ax-ext 2005 |
This theorem depends on definitions: df-bi 110 df-3an 875 df-tru 1231 df-nf 1330 df-sb 1629 df-clab 2010 df-cleq 2016 df-clel 2019 df-nfc 2150 df-rex 2289 df-v 2536 df-sbc 2741 df-csb 2829 df-sn 3355 df-uni 3554 df-iota 4792 df-riota 5391 |
This theorem is referenced by: (None) |
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