![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sbi1v | GIF version |
Description: Forward direction of sbimv 1870. (Contributed by Jim Kingdon, 25-Dec-2017.) |
Ref | Expression |
---|---|
sbi1v | ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6 1863 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | sb6 1863 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) | |
3 | ax-2 7 | . . . . 5 ⊢ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓))) | |
4 | 3 | al2imi 1435 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓))) |
5 | sb2 1744 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) → [𝑦 / 𝑥]𝜓) | |
6 | 4, 5 | syl6 33 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜓)) |
7 | 2, 6 | sylbi 120 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜓)) |
8 | 1, 7 | syl5bi 151 | 1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1330 [wsb 1739 |
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-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-11 1483 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 |
This theorem depends on definitions: df-bi 116 df-sb 1740 |
This theorem is referenced by: sbimv 1870 |
Copyright terms: Public domain | W3C validator |