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Mirrors > Home > ILE Home > Th. List > rspcedvd | GIF version |
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2797. (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 2797 | . 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 1332 ∈ wcel 1481 ∃wrex 2418 |
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 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-v 2691 |
This theorem is referenced by: rspcime 2800 rspcedeq1vd 2802 rspcedeq2vd 2803 updjud 6975 elpq 9467 modqmuladd 10170 modqmuladdnn0 10172 modfzo0difsn 10199 negfi 11031 divconjdvds 11583 2tp1odd 11617 dfgcd2 11738 qredeu 11814 pw2dvdslemn 11879 xmettx 12718 |
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