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Mirrors > Home > ILE Home > Th. List > rspcedv | GIF version |
Description: Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
rspcdv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
rspcdv.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rspcedv | ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcdv.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | rspcdv.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 2 | biimprd 158 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) |
4 | 1, 3 | rspcimedv 2843 | 1 ⊢ (𝜑 → (𝜒 → ∃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 2739 |
This theorem is referenced by: rspcedvd 2847 rexxfrd 4463 enomnilem 7135 enmkvlem 7158 ltexnqq 7406 halfnqq 7408 ltbtwnnqq 7413 genpml 7515 genpmu 7516 genprndl 7519 genprndu 7520 axarch 7889 apreap 8542 |
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