Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sbcthdv | GIF version |
Description: Deduction version of sbcth 2828. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
sbcthdv.1 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
sbcthdv | ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcthdv.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | 1 | alrimiv 1795 | . 2 ⊢ (𝜑 → ∀𝑥𝜓) |
3 | spsbc 2826 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜓 → [𝐴 / 𝑥]𝜓)) | |
4 | 2, 3 | mpan9 275 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∀wal 1282 ∈ wcel 1433 [wsbc 2815 |
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-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-v 2603 df-sbc 2816 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |