![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sbcom | GIF version |
Description: A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.) |
Ref | Expression |
---|---|
sbcom | ⊢ ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco3 1896 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑥 / 𝑧]𝜑) | |
2 | sbcocom 1892 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑) | |
3 | sbcocom 1892 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑) | |
4 | 1, 2, 3 | 3bitr3i 208 | 1 ⊢ ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 [wsb 1692 |
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-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 |
This theorem is referenced by: sb9 1903 |
Copyright terms: Public domain | W3C validator |