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Mirrors > Home > ILE Home > Th. List > csbabg | GIF version |
Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
csbabg | ⊢ (A ∈ 𝑉 → ⦋A / x⦌{y ∣ φ} = {y ∣ [A / x]φ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbccom 2827 | . . . 4 ⊢ ([z / y][A / x]φ ↔ [A / x][z / y]φ) | |
2 | df-clab 2024 | . . . . 5 ⊢ (z ∈ {y ∣ [A / x]φ} ↔ [z / y][A / x]φ) | |
3 | sbsbc 2762 | . . . . 5 ⊢ ([z / y][A / x]φ ↔ [z / y][A / x]φ) | |
4 | 2, 3 | bitri 173 | . . . 4 ⊢ (z ∈ {y ∣ [A / x]φ} ↔ [z / y][A / x]φ) |
5 | df-clab 2024 | . . . . . 6 ⊢ (z ∈ {y ∣ φ} ↔ [z / y]φ) | |
6 | sbsbc 2762 | . . . . . 6 ⊢ ([z / y]φ ↔ [z / y]φ) | |
7 | 5, 6 | bitri 173 | . . . . 5 ⊢ (z ∈ {y ∣ φ} ↔ [z / y]φ) |
8 | 7 | sbcbii 2812 | . . . 4 ⊢ ([A / x]z ∈ {y ∣ φ} ↔ [A / x][z / y]φ) |
9 | 1, 4, 8 | 3bitr4i 201 | . . 3 ⊢ (z ∈ {y ∣ [A / x]φ} ↔ [A / x]z ∈ {y ∣ φ}) |
10 | sbcel2g 2865 | . . 3 ⊢ (A ∈ 𝑉 → ([A / x]z ∈ {y ∣ φ} ↔ z ∈ ⦋A / x⦌{y ∣ φ})) | |
11 | 9, 10 | syl5rbb 182 | . 2 ⊢ (A ∈ 𝑉 → (z ∈ ⦋A / x⦌{y ∣ φ} ↔ z ∈ {y ∣ [A / x]φ})) |
12 | 11 | eqrdv 2035 | 1 ⊢ (A ∈ 𝑉 → ⦋A / x⦌{y ∣ φ} = {y ∣ [A / x]φ}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 [wsb 1642 {cab 2023 [wsbc 2758 ⦋csb 2846 |
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-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 |
This theorem is referenced by: csbsng 3422 csbunig 3579 csbxpg 4364 csbdmg 4472 csbrng 4725 |
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