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Mirrors > Home > ILE Home > Th. List > abexssex | GIF version |
Description: Existence of a class abstraction with an existentially quantified expression. Both x and y can be free in φ. (Contributed by NM, 29-Jul-2006.) |
Ref | Expression |
---|---|
abrexex2.1 | ⊢ A ∈ V |
abrexex2.2 | ⊢ {y ∣ φ} ∈ V |
Ref | Expression |
---|---|
abexssex | ⊢ {y ∣ ∃x(x ⊆ A ∧ φ)} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2306 | . . . 4 ⊢ (∃x ∈ 𝒫 Aφ ↔ ∃x(x ∈ 𝒫 A ∧ φ)) | |
2 | selpw 3358 | . . . . . 6 ⊢ (x ∈ 𝒫 A ↔ x ⊆ A) | |
3 | 2 | anbi1i 431 | . . . . 5 ⊢ ((x ∈ 𝒫 A ∧ φ) ↔ (x ⊆ A ∧ φ)) |
4 | 3 | exbii 1493 | . . . 4 ⊢ (∃x(x ∈ 𝒫 A ∧ φ) ↔ ∃x(x ⊆ A ∧ φ)) |
5 | 1, 4 | bitri 173 | . . 3 ⊢ (∃x ∈ 𝒫 Aφ ↔ ∃x(x ⊆ A ∧ φ)) |
6 | 5 | abbii 2150 | . 2 ⊢ {y ∣ ∃x ∈ 𝒫 Aφ} = {y ∣ ∃x(x ⊆ A ∧ φ)} |
7 | abrexex2.1 | . . . 4 ⊢ A ∈ V | |
8 | 7 | pwex 3923 | . . 3 ⊢ 𝒫 A ∈ V |
9 | abrexex2.2 | . . 3 ⊢ {y ∣ φ} ∈ V | |
10 | 8, 9 | abrexex2 5693 | . 2 ⊢ {y ∣ ∃x ∈ 𝒫 Aφ} ∈ V |
11 | 6, 10 | eqeltrri 2108 | 1 ⊢ {y ∣ ∃x(x ⊆ A ∧ φ)} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∃wex 1378 ∈ wcel 1390 {cab 2023 ∃wrex 2301 Vcvv 2551 ⊆ wss 2911 𝒫 cpw 3351 |
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-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 |
This theorem is referenced by: (None) |
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