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Mirrors > Home > MPE Home > Th. List > sbco2v | Structured version Visualization version GIF version |
Description: A composition law for substitution. Version of sbco2 2506 with disjoint variable conditions but not requiring ax-13 2367. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 29-Apr-2023.) |
Ref | Expression |
---|---|
sbco2v.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
sbco2v | ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco2v.1 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | nfsbv 2319 | . 2 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
3 | sbequ 2079 | . 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 2, 3 | sbiev 2304 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 Ⅎwnf 1778 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-11 2147 ax-12 2167 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-nf 1779 df-sb 2061 |
This theorem is referenced by: cbvabwOLD 2803 clelsb1fw 2903 ichbi12i 46800 |
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