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Mirrors > Home > ILE Home > Th. List > spcedv | GIF version |
Description: Existential specialization, using implicit substitution, deduction version. (Contributed by RP, 12-Aug-2020.) |
Ref | Expression |
---|---|
spcedv.1 | ⊢ (𝜑 → 𝑋 ∈ V) |
spcedv.2 | ⊢ (𝜑 → 𝜒) |
spcedv.3 | ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
spcedv | ⊢ (𝜑 → ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcedv.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ V) | |
2 | spcedv.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | spcedv.3 | . . 3 ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒)) | |
4 | 3 | spcegv 2797 | . 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 1332 ∃wex 1469 ∈ wcel 2125 Vcvv 2709 |
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 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-ext 2136 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1740 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-v 2711 |
This theorem is referenced by: fprodseq 11457 |
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