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Mirrors > Home > ILE Home > Th. List > sbcocom | GIF version |
Description: Relationship between composition and commutativity for substitution. (Contributed by Jim Kingdon, 28-Feb-2018.) |
Ref | Expression |
---|---|
sbcocom | ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsb1 1716 | . . 3 ⊢ [𝑧 / 𝑦]𝑦 = 𝑧 | |
2 | sbequ 1769 | . . . 4 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
3 | 2 | sbimi 1695 | . . 3 ⊢ ([𝑧 / 𝑦]𝑦 = 𝑧 → [𝑧 / 𝑦]([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)) |
4 | 1, 3 | ax-mp 7 | . 2 ⊢ [𝑧 / 𝑦]([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) |
5 | sbbi 1882 | . 2 ⊢ ([𝑧 / 𝑦]([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) ↔ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)) | |
6 | 4, 5 | mpbi 144 | 1 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 [wsb 1693 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 |
This theorem depends on definitions: df-bi 116 df-nf 1396 df-sb 1694 |
This theorem is referenced by: sbcomv 1894 sbco3xzyz 1896 sbcom 1898 |
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