![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sbn | GIF version |
Description: Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.) |
Ref | Expression |
---|---|
sbn | ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbnv 1861 | . . . 4 ⊢ ([𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑥]𝜑) | |
2 | 1 | sbbii 1739 | . . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ [𝑦 / 𝑧] ¬ [𝑧 / 𝑥]𝜑) |
3 | sbnv 1861 | . . 3 ⊢ ([𝑦 / 𝑧] ¬ [𝑧 / 𝑥]𝜑 ↔ ¬ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑) | |
4 | 2, 3 | bitri 183 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑) |
5 | ax-17 1507 | . . . 4 ⊢ (𝜑 → ∀𝑧𝜑) | |
6 | 5 | hbn 1633 | . . 3 ⊢ (¬ 𝜑 → ∀𝑧 ¬ 𝜑) |
7 | 6 | sbco2vh 1919 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥] ¬ 𝜑 ↔ [𝑦 / 𝑥] ¬ 𝜑) |
8 | 5 | sbco2vh 1919 | . . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
9 | 8 | notbii 658 | . 2 ⊢ (¬ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
10 | 4, 7, 9 | 3bitr3i 209 | 1 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 104 [wsb 1736 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 |
This theorem is referenced by: sbcng 2953 difab 3350 rabeq0 3397 abeq0 3398 ssfirab 6830 |
Copyright terms: Public domain | W3C validator |