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Mirrors > Home > MPE Home > Th. List > sbco | Structured version Visualization version GIF version |
Description: A composition law for substitution. See sbcov 2248 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) |
Ref | Expression |
---|---|
sbco | ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcom3 2541 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑦]𝜑) | |
2 | sbid 2247 | . . 3 ⊢ ([𝑦 / 𝑦]𝜑 ↔ 𝜑) | |
3 | 2 | sbbii 2072 | . 2 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
4 | 1, 3 | bitri 276 | 1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 |
This theorem is referenced by: sbid2 2543 sbco3 2548 |
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