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Mirrors > Home > MPE Home > Th. List > sbco2vv | Structured version Visualization version GIF version |
Description: A composition law for substitution. Version of sbco2 2514 with disjoint variable conditions and fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised 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 2089 | . 2 ⊢ (𝑧 = 𝑤 → ([𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)) | |
2 | sbequ 2089 | . 2 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
3 | 1, 2 | sbievw2 2103 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]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 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-sb 2071 |
This theorem is referenced by: sbco4lem 2277 sbco4lemOLD 2278 sbco4 2279 cbvabv 2811 sbralie 3381 sbccow 3717 wl-equsb3 35448 2reu8i 44277 |
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