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Mirrors > Home > ILE Home > Th. List > sbco3 | GIF version |
Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.) |
Ref | Expression |
---|---|
sbco3 | ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco3xzyz 1944 | . . 3 ⊢ ([𝑤 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑥][𝑥 / 𝑦]𝜑) | |
2 | 1 | sbbii 1738 | . 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑤][𝑤 / 𝑥][𝑥 / 𝑦]𝜑) |
3 | ax-17 1506 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑤[𝑦 / 𝑥]𝜑) | |
4 | 3 | sbco2h 1935 | . 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑) |
5 | ax-17 1506 | . . 3 ⊢ ([𝑥 / 𝑦]𝜑 → ∀𝑤[𝑥 / 𝑦]𝜑) | |
6 | 5 | sbco2h 1935 | . 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑) |
7 | 2, 4, 6 | 3bitr3i 209 | 1 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 [wsb 1735 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 |
This theorem is referenced by: sbcom 1946 |
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