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Mirrors > Home > ILE Home > Th. List > spcimedv | GIF version |
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
spcimdv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
spcimedv.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) |
Ref | Expression |
---|---|
spcimedv | ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcimedv.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) | |
2 | 1 | ex 114 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜒 → 𝜓))) |
3 | 2 | alrimiv 1847 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜒 → 𝜓))) |
4 | spcimdv.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
5 | nfv 1509 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
6 | nfcv 2282 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
7 | 5, 6 | spcimegft 2767 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜒 → 𝜓)) → (𝐴 ∈ 𝐵 → (𝜒 → ∃𝑥𝜓))) |
8 | 3, 4, 7 | sylc 62 | 1 ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1330 = wceq 1332 ∃wex 1469 ∈ wcel 1481 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 |
This theorem is referenced by: rspcimedv 2795 fihashf1rn 10567 |
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