![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sbieh | GIF version |
Description: Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1801 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbieh.1 | ⊢ (𝜓 → ∀𝑥𝜓) |
sbieh.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbieh | ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . 2 ⊢ (𝜑 → 𝜑) | |
2 | 1 | hbth 1473 | . . 3 ⊢ ((𝜑 → 𝜑) → ∀𝑥(𝜑 → 𝜑)) |
3 | sbieh.1 | . . . 4 ⊢ (𝜓 → ∀𝑥𝜓) | |
4 | 3 | a1i 9 | . . 3 ⊢ ((𝜑 → 𝜑) → (𝜓 → ∀𝑥𝜓)) |
5 | sbieh.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 5 | a1i 9 | . . 3 ⊢ ((𝜑 → 𝜑) → (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))) |
7 | 2, 4, 6 | sbiedh 1797 | . 2 ⊢ ((𝜑 → 𝜑) → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) |
8 | 1, 7 | ax-mp 5 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1361 [wsb 1772 |
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-5 1457 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-4 1520 ax-i9 1540 ax-ial 1544 |
This theorem depends on definitions: df-bi 117 df-sb 1773 |
This theorem is referenced by: sbie 1801 sbco2vlem 1955 equsb3lem 1961 sbco2yz 1974 dvelimf 2026 elsb1 2166 elsb2 2167 |
Copyright terms: Public domain | W3C validator |