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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 2553 with disjoint variable conditions, not requiring ax-13 2390, but ax-11 2161. (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 2349 | . 2 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
3 | sbequ 2090 | . 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 2, 3 | sbiev 2330 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 Ⅎwnf 1784 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: cbvabw 2892 clelsb3fw 2983 ichbi12i 43625 |
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