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Mirrors > Home > MPE Home > Th. List > sbco4lem | Structured version Visualization version GIF version |
Description: Lemma for sbco4 2279. It replaces the temporary variable 𝑣 with another temporary variable 𝑤. (Contributed by Jim Kingdon, 26-Sep-2018.) (Proof shortened by Wolf Lammen, 12-Oct-2024.) |
Ref | Expression |
---|---|
sbco4lem | ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcom2 2165 | . . 3 ⊢ ([𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑣 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑) | |
2 | 1 | sbbii 2082 | . 2 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑣 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑) |
3 | sbco2vv 2104 | . . 3 ⊢ ([𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑) | |
4 | 3 | 2sbbii 2083 | . 2 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑) |
5 | sbco2vv 2104 | . 2 ⊢ ([𝑥 / 𝑣][𝑣 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑) | |
6 | 2, 4, 5 | 3bitr3i 304 | 1 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 [wsb 2070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-11 2158 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-sb 2071 |
This theorem is referenced by: sbco4 2279 |
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