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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 1785 | . . 3 ⊢ [𝑧 / 𝑦]𝑦 = 𝑧 | |
2 | sbequ 1840 | . . . 4 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
3 | 2 | sbimi 1764 | . . 3 ⊢ ([𝑧 / 𝑦]𝑦 = 𝑧 → [𝑧 / 𝑦]([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ [𝑧 / 𝑦]([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) |
5 | sbbi 1959 | . 2 ⊢ ([𝑧 / 𝑦]([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) ↔ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)) | |
6 | 4, 5 | mpbi 145 | 1 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 [wsb 1762 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 |
This theorem is referenced by: sbcomv 1971 sbco3xzyz 1973 sbcom 1975 |
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