Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > hbsbv | GIF version |
Description: This is a version of hbsb 1823 with an extra distinct variable constraint, on 𝑧 and 𝑥. (Contributed by Jim Kingdon, 25-Dec-2017.) |
Ref | Expression |
---|---|
hbsbv.1 | ⊢ (𝜑 → ∀𝑧𝜑) |
Ref | Expression |
---|---|
hbsbv | ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sb 1646 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
2 | ax-17 1419 | . . . 4 ⊢ (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) | |
3 | hbsbv.1 | . . . 4 ⊢ (𝜑 → ∀𝑧𝜑) | |
4 | 2, 3 | hbim 1437 | . . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) → ∀𝑧(𝑥 = 𝑦 → 𝜑)) |
5 | 2, 3 | hban 1439 | . . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑧(𝑥 = 𝑦 ∧ 𝜑)) |
6 | 5 | hbex 1527 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑧∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
7 | 4, 6 | hban 1439 | . 2 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → ∀𝑧((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
8 | 1, 7 | hbxfrbi 1361 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 ∃wex 1381 [wsb 1645 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-sb 1646 |
This theorem is referenced by: sbco2vlem 1820 2sb5rf 1865 2sb6rf 1866 |
Copyright terms: Public domain | W3C validator |