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| Mirrors > Home > MPE Home > Th. List > rspcimdv | Structured version Visualization version 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 3062 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓)) | |
| 2 | rspcimdv.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
| 3 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) | |
| 4 | 3 | eleq1d 2826 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
| 5 | 4 | biimprd 248 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐴 ∈ 𝐵 → 𝑥 ∈ 𝐵)) |
| 6 | rspcimdv.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) | |
| 7 | 5, 6 | imim12d 81 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐵 → 𝜓) → (𝐴 ∈ 𝐵 → 𝜒))) |
| 8 | 2, 7 | spcimdv 3593 | . . 3 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜓) → (𝐴 ∈ 𝐵 → 𝜒))) |
| 9 | 2, 8 | mpid 44 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜓) → 𝜒)) |
| 10 | 1, 9 | biimtrid 242 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∀wral 3061 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 |
| This theorem is referenced by: rspcimedv 3613 rspcdv 3614 wrd2ind 14761 mreexd 17685 mreexexlemd 17687 catcocl 17728 catass 17729 moni 17780 subccocl 17890 funcco 17916 fullfo 17959 fthf1 17964 nati 18003 acsfiindd 18598 chpscmat 22848 mpomulcn 24891 friendshipgt3 30417 lmxrge0 33951 funressnfv 47055 cfsetsnfsetf1 47071 |
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