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Mirrors > Home > ILE Home > Th. List > rspcedvd | GIF version |
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2847. (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 2847 | . 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 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2741 |
This theorem is referenced by: rspcime 2850 rspcedeq1vd 2852 rspcedeq2vd 2853 updjud 7083 elpq 9650 modqmuladd 10368 modqmuladdnn0 10370 modfzo0difsn 10397 negfi 11238 divconjdvds 11857 2tp1odd 11891 dfgcd2 12017 qredeu 12099 pw2dvdslemn 12167 dvdsprmpweq 12336 oddprmdvds 12354 isnsgrp 12817 dfgrp2 12907 grplrinv 12932 grpidinv 12934 dfgrp3m 12974 ringid 13214 xmettx 14095 bj-charfunbi 14648 |
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