Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > rspcda | GIF version |
Description: Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 29-Jun-2020.) |
Ref | Expression |
---|---|
rspcdva.1 | ⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜒)) |
rspcdva.2 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) |
rspcdva.3 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
rspcda.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
rspcda | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcdva.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
2 | rspcdva.2 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) | |
3 | rspcdva.1 | . . 3 ⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜒)) | |
4 | 3 | rspcv 2697 | . 2 ⊢ (𝐶 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
5 | 1, 2, 4 | sylc 61 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 Ⅎwnf 1389 ∈ wcel 1433 ∀wral 2348 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |