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Mirrors > Home > ILE Home > Th. List > inab | GIF version |
Description: Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
inab | ⊢ ({x ∣ φ} ∩ {x ∣ ψ}) = {x ∣ (φ ∧ ψ)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sban 1826 | . . 3 ⊢ ([y / x](φ ∧ ψ) ↔ ([y / x]φ ∧ [y / x]ψ)) | |
2 | df-clab 2024 | . . 3 ⊢ (y ∈ {x ∣ (φ ∧ ψ)} ↔ [y / x](φ ∧ ψ)) | |
3 | df-clab 2024 | . . . 4 ⊢ (y ∈ {x ∣ φ} ↔ [y / x]φ) | |
4 | df-clab 2024 | . . . 4 ⊢ (y ∈ {x ∣ ψ} ↔ [y / x]ψ) | |
5 | 3, 4 | anbi12i 433 | . . 3 ⊢ ((y ∈ {x ∣ φ} ∧ y ∈ {x ∣ ψ}) ↔ ([y / x]φ ∧ [y / x]ψ)) |
6 | 1, 2, 5 | 3bitr4ri 202 | . 2 ⊢ ((y ∈ {x ∣ φ} ∧ y ∈ {x ∣ ψ}) ↔ y ∈ {x ∣ (φ ∧ ψ)}) |
7 | 6 | ineqri 3124 | 1 ⊢ ({x ∣ φ} ∩ {x ∣ ψ}) = {x ∣ (φ ∧ ψ)} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1242 ∈ wcel 1390 [wsb 1642 {cab 2023 ∩ cin 2910 |
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-in 2918 |
This theorem is referenced by: inrab 3203 inrab2 3204 dfrab2 3206 dfrab3 3207 imainlem 4923 imain 4924 |
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