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Mirrors > Home > ILE Home > Th. List > rspcimedv | GIF version |
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
rspcimdv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
rspcimedv.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) |
Ref | Expression |
---|---|
rspcimedv | ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcimdv.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | simpr 109 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) | |
3 | 2 | eleq1d 2208 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
4 | 3 | biimprd 157 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐴 ∈ 𝐵 → 𝑥 ∈ 𝐵)) |
5 | rspcimedv.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) | |
6 | 4, 5 | anim12d 333 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐴 ∈ 𝐵 ∧ 𝜒) → (𝑥 ∈ 𝐵 ∧ 𝜓))) |
7 | 1, 6 | spcimedv 2772 | . . 3 ⊢ (𝜑 → ((𝐴 ∈ 𝐵 ∧ 𝜒) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))) |
8 | 1, 7 | mpand 425 | . 2 ⊢ (𝜑 → (𝜒 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))) |
9 | df-rex 2422 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) | |
10 | 8, 9 | syl6ibr 161 | 1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∃wrex 2417 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 |
This theorem is referenced by: rspcedv 2793 |
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