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Mirrors > Home > ILE Home > Th. List > rspcimdv | GIF version |
Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
rspcimdv.1 | ⊢ (φ → A ∈ B) |
rspcimdv.2 | ⊢ ((φ ∧ x = A) → (ψ → χ)) |
Ref | Expression |
---|---|
rspcimdv | ⊢ (φ → (∀x ∈ B ψ → χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2305 | . 2 ⊢ (∀x ∈ B ψ ↔ ∀x(x ∈ B → ψ)) | |
2 | rspcimdv.1 | . . 3 ⊢ (φ → A ∈ B) | |
3 | simpr 103 | . . . . . . 7 ⊢ ((φ ∧ x = A) → x = A) | |
4 | 3 | eleq1d 2103 | . . . . . 6 ⊢ ((φ ∧ x = A) → (x ∈ B ↔ A ∈ B)) |
5 | 4 | biimprd 147 | . . . . 5 ⊢ ((φ ∧ x = A) → (A ∈ B → x ∈ B)) |
6 | rspcimdv.2 | . . . . 5 ⊢ ((φ ∧ x = A) → (ψ → χ)) | |
7 | 5, 6 | imim12d 68 | . . . 4 ⊢ ((φ ∧ x = A) → ((x ∈ B → ψ) → (A ∈ B → χ))) |
8 | 2, 7 | spcimdv 2631 | . . 3 ⊢ (φ → (∀x(x ∈ B → ψ) → (A ∈ B → χ))) |
9 | 2, 8 | mpid 37 | . 2 ⊢ (φ → (∀x(x ∈ B → ψ) → χ)) |
10 | 1, 9 | syl5bi 141 | 1 ⊢ (φ → (∀x ∈ B ψ → χ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1240 = wceq 1242 ∈ wcel 1390 ∀wral 2300 |
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-ral 2305 df-v 2553 |
This theorem is referenced by: rspcdv 2653 |
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