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Mirrors > Home > MPE Home > Th. List > sbco | Structured version Visualization version GIF version |
Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2366. See sbcov 2243 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.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbco | ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcom3 2500 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑦]𝜑) | |
2 | sbid 2242 | . . 3 ⊢ ([𝑦 / 𝑦]𝜑 ↔ 𝜑) | |
3 | 2 | sbbii 2071 | . 2 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
4 | 1, 3 | bitri 274 | 1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsb 2059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-12 2166 ax-13 2366 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ex 1774 df-nf 1778 df-sb 2060 |
This theorem is referenced by: sbid2 2502 sbco3 2507 |
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