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Mirrors > Home > MPE Home > Th. List > sbcov | Structured version Visualization version GIF version |
Description: A composition law for substitution. Version of sbco 2510 with a disjoint variable condition using fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Gino Giotto, 7-Aug-2023.) |
Ref | Expression |
---|---|
sbcov | ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcom3vv 2102 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑦]𝜑) | |
2 | sbid 2253 | . . 3 ⊢ ([𝑦 / 𝑦]𝜑 ↔ 𝜑) | |
3 | 2 | sbbii 2082 | . 2 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
4 | 1, 3 | bitri 278 | 1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]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 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-sb 2071 |
This theorem is referenced by: sb6a 2255 |
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