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Mirrors > Home > ILE Home > Th. List > rspcedvd | GIF version |
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2788. (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 2788 | . 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 1331 ∈ wcel 1480 ∃wrex 2415 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-v 2683 |
This theorem is referenced by: rspcime 2791 rspcedeq1vd 2793 rspcedeq2vd 2794 updjud 6960 modqmuladd 10132 modqmuladdnn0 10134 modfzo0difsn 10161 negfi 10992 divconjdvds 11536 2tp1odd 11570 dfgcd2 11691 qredeu 11767 pw2dvdslemn 11832 xmettx 12668 |
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