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Mirrors > Home > MPE Home > Th. List > sbco2vv | Structured version Visualization version GIF version |
Description: Version of sbco2 2477 with disjoint variable conditions and fewer axioms. (Contributed by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 29-Apr-2023.) |
Ref | Expression |
---|---|
sbco2vv | ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ 2035 | . 2 ⊢ (𝑧 = 𝑤 → ([𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)) | |
2 | sbequ 2035 | . 2 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
3 | 1, 2 | sbievw2 2043 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 [wsb 2015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1743 df-sb 2016 |
This theorem is referenced by: sbid2vw 2186 sbco4lem 2412 sbco4 2413 clelsb3vOLD 2895 cbvabv 2911 sbralie 3398 wl-equsb3 34230 wl-dfrabv 34300 wl-dfrabf 34302 2reu8i 42716 |
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