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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 2447 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓)) | |
2 | rspcimdv.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
3 | simpr 109 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) | |
4 | 3 | eleq1d 2233 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
5 | 4 | biimprd 157 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐴 ∈ 𝐵 → 𝑥 ∈ 𝐵)) |
6 | rspcimdv.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) | |
7 | 5, 6 | imim12d 74 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐵 → 𝜓) → (𝐴 ∈ 𝐵 → 𝜒))) |
8 | 2, 7 | spcimdv 2805 | . . 3 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜓) → (𝐴 ∈ 𝐵 → 𝜒))) |
9 | 2, 8 | mpid 42 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜓) → 𝜒)) |
10 | 1, 9 | syl5bi 151 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1340 = wceq 1342 ∈ wcel 2135 ∀wral 2442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 |
This theorem is referenced by: rspcdv 2828 |
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