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Mirrors > Home > ILE Home > Th. List > elabd | GIF version |
Description: Explicit demonstration the class {𝑥 ∣ 𝜓} is not empty by the example 𝑋. (Contributed by RP, 12-Aug-2020.) |
Ref | Expression |
---|---|
elab.xex | ⊢ (𝜑 → 𝑋 ∈ V) |
elab.xmaj | ⊢ (𝜑 → 𝜒) |
elab.xsub | ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
elabd | ⊢ (𝜑 → ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elab.xex | . 2 ⊢ (𝜑 → 𝑋 ∈ V) | |
2 | elab.xmaj | . 2 ⊢ (𝜑 → 𝜒) | |
3 | elab.xsub | . . 3 ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒)) | |
4 | 3 | spcegv 2774 | . 2 ⊢ (𝑋 ∈ V → (𝜒 → ∃𝑥𝜓)) |
5 | 1, 2, 4 | sylc 62 | 1 ⊢ (𝜑 → ∃𝑥𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Vcvv 2686 |
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-v 2688 |
This theorem is referenced by: ntrivcvgap0 11321 |
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