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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 1854 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜒 → 𝜓))) |
4 | spcimdv.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
5 | nfv 1508 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
6 | nfcv 2299 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
7 | 5, 6 | spcimegft 2790 | . 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 1333 = wceq 1335 ∃wex 1472 ∈ wcel 2128 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 |
This theorem is referenced by: rspcimedv 2818 fihashf1rn 10667 |
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