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Mirrors > Home > ILE Home > Th. List > rspcedvd | GIF version |
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2727. (Contributed by AV, 27-Nov-2019.) |
Ref | Expression |
---|---|
rspcedvd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
rspcedvd.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
rspcedvd.3 | ⊢ (𝜑 → 𝜒) |
Ref | Expression |
---|---|
rspcedvd | ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcedvd.3 | . 2 ⊢ (𝜑 → 𝜒) | |
2 | rspcedvd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
3 | rspcedvd.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
4 | 2, 3 | rspcedv 2727 | . 2 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
5 | 1, 4 | mpd 13 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1290 ∈ wcel 1439 ∃wrex 2361 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-rex 2366 df-v 2622 |
This theorem is referenced by: rspcedeq1vd 2731 rspcedeq2vd 2732 updjud 6827 modqmuladd 9834 modqmuladdnn0 9836 modfzo0difsn 9863 negfi 10720 divconjdvds 11189 2tp1odd 11223 dfgcd2 11342 qredeu 11418 pw2dvdslemn 11482 |
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